THE  LIBRARY 

OF 

THE  UNIVERSITY 
OF  CALIFORNIA 

LOS  ANGELES 


THE  INTERFEROMETRY  OF  REVERSED  AND 
NON-REVERSED  SPECTRA 


BY  CARL  BARUS 

Hazard  Professor  of  Physics  and  Dean  of  the  Graduate  Department 
in  Brown  University 


PUBLISHED  BY  THE  CARNEGIE  INSTITUTION  OF  WASHINGTON 
WASHINGTON,  1916 


86443 


CARNEGIE  INSTITUTION  OF  WASHINGTON 
PUBLICATION  No,  249 .,  y 


PRINTED  BY  J.   B.  LIPPINCOTT  COMPANY 

AT  THE  WASHINGTON  SQUARE  PRESS 

PHILADELPHIA,   U.  S.  A. 


8  V  V  K 


SEg 


pt.  /-z 
CONTENTS. 


CHAPTER  I.  —  The  Interferences  of  Crossed  Spectra. 

PAGE 

1.  Introductory  ........  ..................................................... 

2.  Coincident  spectra  with  one  reversed  on  a  given  Fraunhofer  line.     Figs.  I,  2,  3  . 

3.  The  same.    Further  experiments  ...........................................       1  1 

4.  Coincident  spectra  with  one  reversed  on  a  given  longitudinal  axis.     Figs.  4,  5,  6.       12 

5.  Interference  of  the  corresponding  first-order  spectra  of  the  grating,  in  the  absence 


of  rotation.    Figs.  7,  8,  9,  10 
6.  Conclusion 


f  ;y  CHAPTER  II. — Further  Study  of  the  Interference  of  Reversed  Spectra. 

"*         7.  Apparatus  with  one  grating.    Figs.  1 1,  12,  13,  a,  b 19 

8.  Observations  and  experiments  with  a  single  grating.    Fig.  14 22 

9.  Inferences.     Fig.  15,  a,  b 24 

10.  Apparatus  with  two  gratings.    Figs.  16,  17,  18 26 

11.  Experiments  continued.    New  interferometer.    Figs.  19,  20 30 

12.  Experiments  continued.    Homogeneous  light 32 

^    13.  Experiments  continued.    Contrast  of  methods 33 

14.  Experiments  continued.    Rotation,  etc.,  of  grating.    Figs.  21,  22 33 

•v    15.  Tentative  equations.    Figs.  23,  24,  25 36 

16.  Experiments  continued.    Analogies.    Figs.  26,  27 38 

f>    17.  Subsidiary  diffractions.    Figs.  28,  29 43 

*   18.  Conclusion 45 

CHAPTER  III. — The  Interferences  of  Non-reversed  Spectra  of  Two  Gratings. 

19.  Introduction.    Method.    Figs.  30,  31 46 

20.  White  light.    Colored  fringes.    Tables  i,  2,  3.    Figs.  32,  33,  34,  35 47 

21.  Homogeneous  light.  Wide  slit.  Transverse  axes  coincident.    Tables  4,  5.    Fig.  36..  52 

22.  Homogeneous  light.    Fine  slit.    Transverse  axes  not  coincident.    Table  6.    Fig.  37.  54 
xi1    23.  Homogeneous  light.    Slit  and  collimator  removed.    Table  7.    Fig.  38 55 

^    24.  Inferences.    Figs.  39,  40 56 

^    25.  Rotation  of  colored  fringes.    Non-reversed  spectra.    Figs.  41,  42 58 

^_   26.  Final  treatment  of  reversed  spectra.    Hypothetical  case.    Figs.  43,  44,  45,  46 ....  60 

^    27.  Case  of  reflecting  grating.    Homogeneous  light.    Figs.  47,  48 64 

J    28.  Non-symmetrical  positions.    Fore-and-aft  motion.    Fig.  49 67 

j  CHAPTER  IV. — The  Distance  Between  Two  Parallel  Transparent  Plates. 

5  29.  Introductory 69 

>j   30.  Apparatus.    Figs.  50,  51 69 

i    31.  Equations.    Figs.  52,  53 7° 

*•   32.  Method 72 

v    33.  Observations  and  corrections.    Preliminary  work.    Figs.  54,  55 73 

CHAPTER  V. — Interferometers  for  Parallel  and  for  Crossed  Rays. 

34.  Introduction.    Methods.    Figs.  56,  57 78 

35.  Experiment.    Reflecting  grating.    Parallel  rays.    Fig.  58 79 

36.  Experiments.    Transmitting  grating.    Parallel  rays 8 1 

37.  Experiments.    Transmitting  grating.    Crossed  rays.    Figs.  59,  60,  61,  a,  b 82 

38.  The  same.    The  linear  phenomenon.    Fig.  62 85 

39.  The  same.    Inferences.    Figs.  63,  64,  65 87 

40.  Experiments.    Reflecting  grating.    Crossed  rays.    Figs.  66,  67 88 

41 .  The  same.    Compensators 9 l 

42.  Miscellaneous  experiments.    Fringes  with  mercury  light 91 

43.  Inferences.    Figs.  68,  69 92 

3 


4  CONTENTS. 

CHAPTER  VI. — Channeled  Spectra  Occurring  in  Connection  with  the  Diffraction  of 
Reflecting  Gratings. 

44.  Introductory 95 

45.  Apparatus.    Fig.  70 95 

46.  Scattering 95 

47.  Fringes  with  white  light 96 

48.  Fringes  with  sodium  light 97 

49.  Grating  on  a  spectrometer.   Fig.  71 98 

50.  Inferences 100 

CHAPTER  VII. — Prismatic  Methods  in  Reversed  and  Non-reversed  Spectrum 
Inter ferometry. 

51.  Purpose 102 

52.  Method  and  apparatus.    Figs.  72,  73 102 

53.  The  same.    Crossed  rays 103 

54.  Another  method.    Fig.  74 104 

55.  Methods  using  prismatic  dispersion.    Fig.  75 105 

56.  Methods  with  paired  prisms.    Fig.  76 106 

CHAPTER  VIII. — The  Linear  Type  of  Displacement  Interferometers. 

57.  Introductory 107 

58.  Apparatus.     Fig.  77 107 

59.  Film  grating.    Adjustment.    Figs.  78,  79 109 

60.  Michelson's  interferences no 

61.  Film  grating.    Another  adjustment.    Fig.  80 in 

62.  Equations in 

CHAPTER  IX. — The  Use  of  Compensators  Bounded  by  Curved  Surfaces. 

63.  Introduction 113 

64.  Lens  systems 113 

65.  Effective  thickness  of  the  lenticular  compensator.   Fig.  81 115 

66.  Observations  largely  with  weak  lenses  and  short  interferometer.    Figs.  82,  83 1 16 

67.  Remarks.    Fig.  84 1 18 

68.  Observation  with  lens  systems  on  both  sides.    Figs.  85,  86 1 19 

69.  Telescopic  interferences.    Figs.  87,  88,  89,  90,  91 120 

CHAPTER  X. — The  Dispersion  of  Air. 

70.  Introduction.    Table  8 124 

71.  Observations  with  arc  lamp 124 

72.  Observations  with  sunlight.    Single  tube.    Table  9 125 

73.  Two  (differential)  refraction  tubes.    Table  10.    Fig.  92 127 

74.  Differential  and  single  refraction  tubes.   Sunlight.   Tables  11,12 129 

75.  Distortion  of  glass  absent 131 

76.  Further  observations  with  sunlight.    Table  13 131 

77.  Conclusion 132 

CHAPTER  XI. — The  Refraction  of  Air  with  Temperature. 

78.  Apparatus.    Fig.  93.    Table  14 133 

79.  Observations 134 

80.  Computation 135 

81.  Final  experiments  at  100°.    Table  15 136 

82.  Experiments  at  red  heat 137 

83.  Further  experiments  at  high  temperatures.     Fig.  94.     Table  16 139 

84.  Flames 140 

85.  Conclusion 141 

CHAPTER  XII. — Adiabatic  Expansion  Observed  with  the  Interferometer. 

86.  Introductory.    Table  17 142 

87.  Experiments  with  short,  bulky  air-chambers 143 

88.  Effect  of  strained  glass 145 

89.  Equations 146 

90.  Experiments  with  long  tubes.    Diameter,  i  inch.    Table  18 148 

91.  The  same.    Diameter  of  tube,  2  inches.    Table  19 150 

92.  The  same.    Diameter  of  tube,  4  inches.    Tables  20,  21.    Fig.  95 151 

CHAPTER  XIII. — Miscellaneous  Experiments. 

93.  Effect  of  ionization  on  the  refraction  of  a  gas 154 

94.  Mach's  interferences.    Fig.  96 155 

95.  A  Rowland  spectrometer  for  transmitting  and  reflecting  gratings,  plane  or  concave. 

Figs.  97,  98,  99 156 


PREFACE. 

The  following  account  of  my  experiments  has  been  given  chronologically. 
Although  many  of  the  anomalous  features,  in  which  the  interferences  of 
superposed  coordinated  spectra  first  presented  themselves,  were  largely 
removed  in  the  later  work,  yet  the  methods  used  in  the  several  papers,  early 
and  later,  are  throughout  different.  It  therefore  seemed  justifiable  to  record 
them,  together  with  the  inferences  they  at  first  suggested.  The  pursuit  of 
the  subject  as  a  whole  was  made  both  easier  and  more  difficult  by  the  un- 
avoidable tremors  of  the  laboratory  in  which  I  am  working;  for  it  is  possibly 
easier  to  detect  an  elusive  phenomenon  if  it  is  in  motion  among  other  similar 
stationary  phenomena.  But  it  is  certainly  difficult,  thereafter,  to  describe 
it  when  found. 

It  will  be  convenient  to  refer  to  the  cases  in  which  one  of  the  two  coincident 
spectra  from  the  same  source  is  rotated  180°  with  reference  to  the  other  on 
a  transverse  axis  (i.e.,  an  axis  parallel  to  the  Fraunhofer  lines),  under  the 
term  reversed  spectra;  while  the  term  inverted  spectra  is  at  hand  for  those  cases 
in  which  one  of  the  paired  spectra  is  turned  180°  relative  to  the  other  on  a 
longitudinal  axis  (i.e.,  an  axis  parallel  to  the  r-v  length  of  the  spectrum). 
In  this  book  the  latter  are  merely  touched  upon,  briefly,  in  Chapter  I,  but 
they  are  now  being  investigated  in  detail  and  give  promise  of  many  interest- 
ing results.  The  chapter  contains  a  full  account  of  what  may  be  seen  with 
a  single  grating — the  linear  phenomenon,  as  I  have  called  it,  and  which,  if 
it  stood  alone,  would  be  difficult  to  interpret. 

In  Chapter  II,  therefore,  the  interferences  of  reversed  spectra  are  treated 
by  the  aid  of  two  gratings,  in  virtue  of  which  a  multitude  of  variations  are 
inevitably  introduced.  The  phenomena  are  thus  exhibited  in  a  way  leading 
much  more  smoothly  to  their  identification. 

This  endeavor  is  given  greater  promise  in  Chapter  III,  which  contains  a 
comparison  of  the  interferences  of  reversed  and  non-reversed  spectra,  the 
latter  produced  in  a  way  quite  different  from  those  in  my  earlier  work.  Nat- 
urally these  in  their  entirety  are  even  more  bewilderingly  varied,  and  become 
particularly  so  when,  as  in  Chapter  IV,  an  intermediate  reflection  of  one 
spectrum  is  admitted.  But  with  this  I  was  on  more  familiar  ground,  as  I 
have  hitherto,  in  these  publications,  given  such  investigations  particular 
attention. 

The  flexibility  of  the  new  methods  is  well  shown  in  Chapter  V,  where 
separated  component  beams  can  with  equal  facility  be  made  to  run  in  parallel, 
or  across  each  other  at  any  angle,  and  perhaps  both,  with  the  double  result 
visible  in  the  field  of  the  telescope.  In  case  of  crossed  rays  a  remarkable 
phenomenon  is  shown,  in  which  very  small  differences  in  wave-length  imply 
a  remarkably  large  difference  in  rotational  phase  (virtually  resolving  power) 
of  the  two  interesting  groups  of  interference  fringes  due  to  each  wave-length. 

5 


6  PREFACE. 

Spectra  obtained  with  two,  or  at  times  even  with  one  grating,  are  often 
annoyingly  furrowed  with  large  transverse  fringes.  These  are  investigated  in 
Chapter  VI,  and  referred  to  diffractions  resulting  from  residual  errors  in  the 
rulings.  In  Chapter  VII,  finally,  several  examples  of  new  methods  of  investi- 
gation are  given.  They  show  the  important  bearing  of  the  diffraction  at 
the  slit  of  the  collimator  on  all  these  experiments.  The  cleavage  of  a  field 
of  diffracted  rays  as  an  essential  preliminary  is  here  put  in  direct  evidence. 

In  Chapters  VIII  to  XIII  I  have  returned  to  my  older  experiments  with 
the  displacement  interferometer.  The  subjects  adduced,  like  the  dispersion 
of  air  at  low  and  high  temperatures,  the  adiabatic  expansion  of  air,  etc.,  are 
pursued  less  with  the  object  of  reaching  results  of  precision  than  of  testing 
the  limits  of  the  displacement  method  and  developing  it. 

My  thanks  are  due  to  Miss  Abbie  L.  Caldwell  for  very  efficient  assistance 
in  preparing  the  manuscript  and  drawings  for  the  press. 

CARL  BARUS. 
BROWN  UNIVERSITY,  Providence,  Rhode  Island. 


CHAPTER  I. 


THE  INTERFERENCES  OF  CROSSED  SPECTRA. 

1.  Introductory. — If  two  component  spectra  from  the  same  source  coincide 
throughout  their  extent  the  elliptic  interferences  will  be  spread  over  the 
whole  surface,  provided,  of  course,  the  respective  glass  and  air-path  differences 
of  the  two  component  rays  are  not  so  great  as  to  throw  the  phenomenon 
beyond  the  range  of  visibility.  In  the  usual  method  of  producing  these 
interferences,  where  the  corresponding  reflections  and  transmissions  of  the 
two  component  rays  take  place  at  the  same  points  of  the  same  plane  surface, 
the  interference  pattern  is  automatically  centered,  or  nearly  so.  This  is  not 
the  case  when,  as  in  the  following  experiments,  the  interfering  beams  are 
separated  in  some  other  way;  and  the  problem  of  centering  is  often  one  of 
the  chief  difficulties  involved;  and  if  the  beams  are  to  be  treated  independ- 
ently, it  is  difficult  to  obviate  this  annoyance. 

Suppose,  now,  that  one  of  the  spectra  is  rotated  around  an  axis  normal  to 
both,  by  a  small  angle.  Will  the  interferences  at  once  vanish,  or  is  there  a 
limiting  angle  below  which  this  is  not  the  case?  In  other  words,  how  far 
can  one  trench  with  light -waves  upon  the  case  of  musical  beats,  or  of  inter- 
ferences not  quite  of  the  same  wave-length? 

Instead  of  approaching  the  question  in  this  form,  in  which  it  would  be 
exceedingly  difficult,  experimentally,  I  have  divided  it  into  two  component 
parts.  Let  one  of  the  spectra  be  rotated  180°  around  a  longitudinal  axis, 
parallel  to  the  red-violet  length  of  the  spectrum  and  normal  to  the  Fraun- 
hofer  lines.  In  such  a  case,  interference  should  be  possible  only  along  the 
infinitely  thin  longitudinal  axis  of  rotation  to  which  both  spectra  are  sym- 
metrical, one  being  the  mirror  image  of  the  other.  One  would  not  expect 
these  interferences  to  be  visible.  It  is  rather  surprising,  however,  that  this 
phenomenon  (as  I  have  found)  may  actually  be  observed,  along  a  definite 
longitudinal  band  in  the  spectrum,  about  twice  the  angular  width  of  the 
distance  between  the  sodium  lines  and  symmetrical  with  respect  to  the  axis 
of  rotation.  It  is  independent  of  the  width  of  the  slit,  provided  this  is  narrow 
enough  to  show  the  Fraunhofer  lines  to  best  advantage. 

Again,  let  one  spectrum  be  rotated  180°  about  a  given  Fraunhofer  line 
(transverse  axis),  the  nickel  or  mean  D  line,  for  instance.  The  two  coplanar 
spectra  are  now  mutually  reversed,  showing  the  succession  red-violet  and 
violet-red,  respectively.  Interference  should  take  place  only  along  the  mean 
D  line  and  be  again  inappreciable.  Experimentally,  I  was  not  at  first  able 
to  find  any  interferences  for  this  case  in  the  manner  shown  below,  but  this 
may  have  been  due  to  inadequacies  in  the  experimental  means  employed, 
for  the  dispersion  was  insufficient  and  the  reflecting  edge  of  the  paired  mirrors 
too  rough.  Improving  the  apparatus,  I  eventually  found  the  phenomenon, 

7 


8 


THE   INTERFEROMETRY   OF 


but  appearing  as  a  single  line,  vividly  colored  above  the  brightness  of  the 
spectrum;  or,  again,  more  jet-black  than  the  Fraunhofer  lines  and  located 
in  the  position  of  the  coincident  wave-lengths  of  the  two  superimposed  spectra. 

It  is  possible,  however,  as  will  be  shown  in  §  4,  to  obtain  two  spectra  in 
such  a  way  that  if  their  longitudinal  axes  coincide  the  Fraunhofer  lines 
intersect  at  a  small  angle,  and  vice  versa.  In  such  a  case,  for  coincident 
Fraunhofer  lines,  interference  occurs  in  a  band  around  these  lines  and  is 
absent  in  the  rest  of  the  spectrum;  whereas,  if  the  longitudinal  axes  are 
coincident,  the  interferences  are  arranged  with  reference  to  these  axes.  These 
results  seem  to  bear  on  the  question,  but  it  is  difficult  to  clearly  resolve  them. 

The  methods  used  in  this  paper  consist  chiefly  in  bringing  the  two  first- 
order  spectra  of  a  grating,  or  the  second-order  spectra  or  their  equivalents, 
to  interfere.  In  this  respect  they  contain  an  additional  method  of  inter- 
ferometry  which  may  be  useful,  if  for  any  reason  it  is  necessary  that  the  two 
component  beams  are  not  to  retrace  their  paths. 


2.  Coincident  spectra  with  one  reversed  on  a  given  Fraunhofer  line. — In 

figure  i,  L  is  a  narrow  vertical  sheet  (subsequently  broadened  by  the  dif- 
fraction of  the  slit)  of  white  sunlight  or  arc  light  from  a  collimator,  G  the 
transparent  grating  ruled  on  the  side  g,  from  which  the  first  or  second  order 
of  spectra  gM  and  gN  originate.  M  and  N  are  opaque  mirrors  mounted 
adjustably  on  a  firm  rail,  RR,  each  of  them  with  three  adjustment  screws 
relative  to  horizontal  and  vertical  axes.  M  is  provided  with  a  slide  micrometer 
(not  shown).  From  M  and  N  the  beams  pass  to  the  smaller  paired  mirrors, 
m  and  n,  which  should  meet  in  a  fine  vertical  line  at  a  very  obtuse  angle. 
A  silvered  biprism  would  here  be  far  preferable,  but  none  having  the  required 
angle  was  available.  From  n,  m,  the  beams  pass  into  the  telescope  T.  As 
the  spectra  are  each  divergent  after  issuing  from  g,  they  can  be  made  to 
overlap  on  leaving  n,  m,  by  aid  of  the  adjustment  screws  on  M  and  N.  More- 
over, as  the  spectra  are  mirror  images  of  each  other,  as  suggested  in  figure  i, 
any  spectrum  lines  (as,  for  instance,  the  D)  may  be  put  in  coincidence  on 
using  one  of  the  adjustment  screws  specified.  It  is  necessary  that  the  telescope 
T  be  sufficiently  near  M  in  order  that  the  micrometer  may  be  manipulated. 


REVERSED   AND   NON-REVERSED   SPECTRA. 


The  D  lines  placed  in  coincidence  are  obviously  opposites,  each  line  being 
paired  with  the  mate  of  the  other.  A  fine  wire  must  be  drawn  across  the  slit 
of  the  collimator,  in  order  that  the  vertical  coincidence  may  be  tested.  One 
should  expect  the  interferences  to  appear  between  the  D  lines  on  gradually 
moving  the  micrometer  mirror  M,  parallel  to  itself,  into  the  required  posi- 
tion. As  stated  above,  I  did  not  at  first  succeed  in  finding  the  interferences, 
but  the  experiment  is  a  delicate  one.  In  a  repetition  with  first-order  spectra, 
it  would  be  advisable  to  replace  the  plane  mirrors  m,  n,  by  slightly  concave 
mirrors,  about  2  meters  in  focal  distance,  and  to  replace  the  telescope  T  by 
a  strong  eyepiece.  This  is  the  method  used  in  the  next  paragraph,  and  it 
was  more  easily  successful. 

Later  I  returned  to  the  experiment  with  the  same  adjustment,  except 
that  the  plane  mirrors  m,  n,  were  placed  beyond  the 
grating,  with  the  object  of  using  the  equivalent  of 
second-order  spectra  to  get  more  dispersion.  This  plan 
did  not  fail,  and,  having  once  obtained  the  interferences, 
the  reproduction  seemed  quite  easy,  as  they  remained 
visible  while  the  micrometer  M  was  moved  over  about 
5  mm.  or  more,  a  very  important  observation.  Their 
appearance  with  a  small  telescope  was  that  of  a  single 
fine  line,  alternately  flaming  yellow  (very  bright  on  the 
yellow  background  of  the  surrounding  part  of  the  spec- 
trum) and  jet  black  as  compared  with  the  D  lines, 
between  which  the  interferential  line  was  situated,  and 
on  an  enhanced  yellow  ground.  The  flicker  is  referable 
to  the  tremor  of  the  laboratory,  which  makes  it  im- 
possible to  keep  these  interferences  quiet.  Shutting  off 
the  light  from  either  mirror,  M  or  N,  naturally  quenches 
the  interferences,  but  leaves  the  yellow  part  of  the 
spectrum  behind. 

Obviously,  coincidence  of  the  longitudinal  axes  of  the 
spectra  alone  is  needed.  Therefore,  upon  moving  the 
two  double  D  lines  apart,  by  aid  of  the  adjustment  screws  on  the  mirror  M 
and  N,  symmetrically  to  the  ends  of  the  yellow  field  in  the  telescope,  the 
interferences  were  isolated  and  located  midway  between  the  D  doublets  of 
each  spectrum,  i.e.,  in  the  center  of  the  field  of  the  telescope.  They  could 
now  be  observed  to  better  advantage.  In  the  small  telescope  there  is  appar- 
ently but  one  dark  line.  If  stationary,  its  ultimate  character,  when  centered, 
would  be  surmised  to  be  given  by  the  intersection  of  a  vertical  diameter  with 
a  series  of  confocal  ellipses,  successively  bright  and  dark,  as  indicated  in 
figure  2.  The  light  and  dark  parts  alternate  or  flicker.  On  moving  the 
micrometer,  the  vertical  intersector  A  takes  a  more  and  more  lateral  position 
like  B,  so  that  the  trembling  interferences  would  soon  be  invisible,  as  they 
rapidly  become  finer  and  hair-like  (not  shown). 

On  using  higher  magnification  (larger  telescope),  two  black  lines  bordering 


Sb 


10  THE   INTERFEROMETRY  OF 

a  bright  line,  or  a  black  line  between  two  bright  lines,  seemed  to  be  visible; 
but  the  interferences  would  have  to  be  stationary  to  be  definitively  described, 
since  the  width  of  the  pattern  is  not  more  than  one-third  to  one-half  of  the 
distance  between  the  sodium  lines.1  The  interferences,  moreover,  did  not 
now  readily  conform  to  the  design  B,  figure  2,  anticipated,  but  were  more  of 
the  type  C,  with  long,  dark  lines  slightly  oblique  to  the  vertical,  and  vibrating 
within  a  vividly  yellow  band.  Sometimes  these  were  heavier,  with  two  or 
three  faint  lines  on  one  side. 

Further  experiment  showed  that  the  phenomenon  is  not  influenced  by 
the  width  of  the  slit,  except  that  it  is  clearest  and  sharpest  with  the  narrowest 
slit  possible  and  vanishes  when  the  slit  is  made  so  wide  that  the  Fraunhofer 
lines  disappear.  It  may  easily  be  produced  by  the  modified  method  following, 
in  any  wave-length  red,  yellow,  green,  etc.,  with  no  essential  difference  except 
in  size.  It  is  present,  moreover,  in  all  focal  planes,  i.e.,  the  ocular  of  the 
telescope  may  be  inserted  or  pulled  out  to  any  distance,  yet  the  same  phe- 
nomena persist  on  the  vague,  colored  background.  A  number  of  observations 
were  made  to  detect  the  change 
of  the  pattern  of  the  interference, 
between  its  entrance  into  the  field 
and  its  eventual  evanescence,  in 
case  of  the  continuous  displace- 
ment of  the  mirror  M  over  5  mm. 
In  figure  2  this  would  be  equiva- 
lent to  a  passage  of  B  into  B' 
through  A,  and  the  fringes  for  a 
distant  center  should  therefore  o 

rotate,  as  they  actually  do  in 
the  experiments  of  the  next  para- 
graph. But  in  the  present  case  the  type  C  persists;  the  lines  may  become 
longer  or  all  but  coalesce  and  their  inclination  may  change  somewhat. 
They  nevertheless  remain  fine  and  nearly  vertical,  until  they  vanish  completely 
and  there  is  no  rotation.  Nor  could  the  phenomenon  be  found  again  within 
the  length  of  the  given  micrometer  screw.  Hence  it  is  improbable  that  these 
interferences  conform  at  once  to  the  ordinary  elliptic  type  for  which  figure  2 
applies,  even  if  the  ellipse  is  considered  exceptionally  eccentric.  The  use  of 
two  slits,  one  following  the  other,  does  not  change  the  pattern. 

The  modified  method  of  experiment  was  one  of  double  diffraction.  In 
figure  3,  L  is  the  blade  of  light  from  the  collimator,  which  passing  under  the 
plane  mirror,  m,  penetrates  the  grating  G,  whence  the  diffracted  first-order 
beams  reach  the  opaque  mirrors  M  and  N.  These  return  the  beams,  nearly 
normally  but  with  an  upward  slant,  so  that  the  color  selected  intersects  the 

lfThe  use  of  the  DI  D2  distance  of  the  sodium  lines  for  the  measurement  of  the  breadth 
of  the  interference  phenomenon  is  a  mere  matter  of  convenience  in  describing  it.  It  will 
be  shown  in  the  next  report  that  the  breadth  of  the  strip  carrying  interference  fringes  is 
quite  independent  of  the  dispersion  of  the  optic  system. 


REVERSED   AND   NON-REVERSED    SPECTRA.  11 

grating  at  a  higher  level  than  L.  A  second  diffraction  takes  place  at  about 
the  same  angle,  9,  to  the  direct  ray  t,  and  the  coincident  rays  now  impinge 
on  the  mirror  m.  They  are  thence  reflected  into  the  telescope  at  T.  This 
method  admits  of  easier  adjustment,  as  everything  is  controlled  by  the  adjust- 
ment screws  on  M  and  N.  Plane  mirrors  M,  N,  and  m  only  are  needed,  the 
latter  being  on  a  horizontal  axis  to  accommodate  T.  The  direct  (white) 
beam  is  screened  off  after  transmission  through  the  grating,  if  necessary. 
But  it  rarely  enters  the  telescope. 

3.  The  same.  Further  experiments. — In  place  of  the  plane  mirror,  m,  a 
slightly  concave  mirror  (2  meters  in  focal  distance,  say)  may  be  used  with 
advantage  and  the  telescope  T  replaced  by  a  strong  eyepiece.  In  this  way 
I  obtained  the  best  results. 

It  is  to  be  noticed  that  the  apparatus  (fig.  3)  may  serve  as  a  spectrometer, 
provided  the  wave-length  X  of  one  line  and  the  grating  space  D  are  known, 
and  the  mirror,  M,  is  measurably  revolvable  about  a  vertical  axis.  In  this 
case  any  unknown  wave-length,  X',  is  obtained  by  rotating  M  until  X'  is  in 
coincidence  with  X.  Supposing  the  X's  of  the  two  spectra  to  have  been  origi- 
nally in  coincidence  and  that  6  is  the  angle  of  M  which  now  puts  X'  in  coin- 
cidence with  X,  it  is  easily  shown  that 

X'-X  =  X  (2  sin2  0/2  +  \/£>2A2-i  sin  0) 

Angles  must  in  such  a  case  be  accurately  measurable,  i.e.,  to  about  o.i  minute 
of  arc  per  Angstrom  unit,  if  the  grating  space  .0  =  351  Xio-6,  as  above. 
Counter-rotation  of  the  mirror  N  till  the  X's  coincide  would  double  the  accu- 
racy. The  usual  grating,  however,  has  greater  dispersion  and  would  require 
less  precision  in  6. 

Finally,  a  still  simpler  and  probably  more  efficient  device  consists  in  com- 
bining the  mirror  m  and  the  plane  grating  G,  or  of  proceeding,  in  other  words, 
on  the  plan  of  Rowland's  method  for  concave  reflecting  gratings.  In  such 
a  case  the  light  would  enter  in  the  direction  TG,  figure  3,  be  reflected  along 
CM ,  back  along  MG,  and  then  return  along  GT  at  a  slightly  higher  or  lower 
level  than  on  entering.  The  equation  just  given  would  still  apply,  and  many 
interesting  modifications  are  suggested.  Experiments  of  this  kind  are  to  be 
tested.  Moreover,  in  case  of  the  plane-transmitting  grating  and  plane 
mirror,  as  above  shown,  the  same  simplification  is  possible  if  the  lens  is 
replaced  by  the  telescope  at  T.  But  in  this  case  the  spectra  are  intersected 
by  strong,  stationary  interferences  due  to  reflections  from  front  and  rear 
faces  and  consequently  not  conveniently  available.  A  reflecting  grating 
and  telescope  would  not  encounter  this  annoyance.  In  general,  however, 
as  in  the  disposition  adopted  in  figure  3,  the  light  enters  opposite  the  observer, 
and,  as  the  light  directly  transmitted  can  be  screened  off,  this  is  a  practical 
convenience  in  favor  of  the  transparent  grating.  The  reflected  spectra  used 
may  be  placed  at  any  level  by  rotating  the  mirror  m  on  a  horizontal  axis. 

On  further  repeating  the  work  by  the  use  of  the  concave  mirror  m,  a  strong 
eyepiece  at  T,  figure  3,  and  using  a  compensator,  I  eventually  succeeded  in 


12  THE   INTERFEROMETRY   OF 

erecting  the  interference  design  C,  figure  2.  It  then  took  the  form  given  at  D, 
and  this  seems  to  furnish  the  final  clue  to  the  subject.  In  other  words,  the 
design  consists  of  a  new  type  of  extremely  eccentric  ellipses,  with  their  long 
axes  parallel  to  the  Fraunhofer  lines,  each  end  having  the  outline  of  a  needle- 
point, possibly  even  concave  outward.  Only  one  end  of  the  long,  closed 
curves  is  obtainable.  These  jet-black  lines  dance  on  the  highly  colored  back- 
ground of  less  than  half  the  width  between  the  two  sodium  lines.  The  inter- 
ference design  would,  therefore,  be  the  same  (apart  from  color)  as  that  which 
would  be  obtained  if  the  spectrum  containing  ordinary  elliptic  interferences 
were  to  shrink  longitudinally  from  red  to  violet,  till  it  occupied  less  than  half 
the  space  between  the  two  D  lines.  In  fact,  I  have  at  other  times  obtained 
just  such  patterns,  with  all  the  colors  present,  but  not  in  the  pure  yellow,  as  in 
the  present  case.  Vertically,  the  path-difference  is  always  due  to  more  or  less 
obliquity  of  the  rays  passing  through  the  plate  of  the  grating.  Horizontally, 
however,  the  equivalent  path-difference  is  complicated,  in  the  present  case,  by 
the  fact  that  one  wave-length  of  a  pair  has  increased,  whereas  the  other  has 
diminished,  while  both  may  pass  through  the  same  thickness  of  glass  and  air. 

4.  Coincident  spectra  with  one  reversed  on  a  given  longitudinal  axis. — For 

this  experiment  it  is  necessary  to  reflect  the  first-order  spectra  issuing  at 
the  grating  G,  figure  4,  from  the  ruled  face  g  (a  narrow,  preferably  horizontal, 
blade  of  white  light  is  here  furnished  by  the  collimator  L  with  a  horizontal 
slit,  and  the  rulings  of  the  grating  are  also  horizontal  and  parallel  to  it) ,  twice 
in  succession  and  preferably  from  mirrors  M  and  N  and  m  and  n,  reflecting 
normally  to  each  other  and  inclined  at  an  angle  of,  roughly,  45°.  Each  of 
the  mirrors  M  and  N  must  be  revolvable  about  a  horizontal  axis  parallel  to 
the  slit  and  furnished  with  three  adjustment  screws  relatively  to  axes  normal 
to  each  other,  one  of  which  is  horizontal.  The  mirrors  m,  n  are  the  silvered 
faces  of  a  prism  right-angled  at  the  edge.  It  is,  moreover,  to  be  placed  on 
the  slide  of  a  Fraunhofer  micrometer  so  that  the  prism  may  be  moved,  grad- 
ually up  and  down,  for  the  adjustment  of  distances. 

On  leaving  the  mirror  m,  n,  the  two  spectra  are  carried  by  nearly  horizontal 
and  parallel  sheets  of  divergent  rays,  which  pass  outward  from  the  diagram. 
But  it  will  be  seen  that  one  of  the  two  spectra  reaching  the  observer  is  reversed 
on  the  longitudinal  axis  relatively  to  the  other;  i.e.,  if  one  is  in  the  position 

1  violet,  the  other  will  be  red  jb°ttOmj  violet, 
j  [ top        j 

The  subsequent  passage  of  the  rays  is  shown  in  figure  5,  which  is  the  side 
elevation  and  therefore  at  right  angles  to  the  preceding  figure.  The  rays 
from  m  and  n  impinge  on  a  distant,  slightly  concave  mirror  K  (about  1.74 
meters  in  focal  distance),  placed  somewhat  obliquely,  so  that  when  the  rays 
come  to  a  focus  at  F  near  the  micrometer  they  may  just  avoid  it .  The  partially 
overlapping  spectra  at  F  are  viewed  by  a  strong  eyepiece,  E.  The  observer 
at  E  can  then  control  the  Fraunhofer  micrometer  by  which  m,  n  is  raised  and 
lowered,  and  the  three  adjustment  screws  of  M. 


REVERSED   AND   NON-REVERSED   SPECTRA. 


13 


The  adjustment  consists  in  first  roughly  placing  all  parts  in  symmetry  with 
sunlight,  until  the  two  spectra  appear  at  E.  The  lens  may  be  removed. 
There  should  be  a  bright,  narrow  spectrum  band  on  each  side  of  and  near  the 
edge  of  the  prism  mn;  for  it  is  clear  that  after  passing  the  lens  E,  correspond- 
ing rays  from  M  and  N  must  both  enter  the  pupil  of  the  eye  to  be  seen  together. 
To  make  the  spectrum  parallel,  the  mirror  mn  is  rotated,  as  a  whole,  around 
a  vertical  axis.  The  three  screws  on  the  mirrors  M  and  N  then  assist  in 
completing  the  adjustment;  the  rotation  around  the  horizontal  axis  brings 


the  sodium  lines  in  coincidence  (both  must  be  clearly  seen  and  sharp  and  at 
an  appreciable  distance  apart);  that  around  the  oblique  axis  gives  rise  to 
more  or  less  overlapping,  as  required.  The  need  of  a  sharp  coincidence  of 
the  sodium  lines  is  very  essential  in  all  these  experiments. 

After  proper  vertical  position  of  mn  has  been  found  by  slowly  moving  the 
micrometer  screw  up  and  down,  the  fringes  appear.  They  are  usually  very 
fine  lines,  possibly  indicating  distant  centers  of  the  ellipses  to  which  they 
belong .  The  appearance  is  roughly  suggested  in  figure  6 .  They  are  thus  totally 
different  from  the  preceding  set,  §  3.  They  pass  from  the  type  a  through  b 


(contraction  toward  the  violet  end  was  not  noticed)  into  the  type  c,  when 
the  mirrors  mn  move  in  a  given  direction.  The  center  of  the  ellipses  is  in 
the  vertical  through  the  field  of  view  for  the  adjustment  b,  in  which  case  the 
lines  pass  from  end  to  end  of  the  spectrum  as  a  narrow  band  near  the  longi- 
tudinal axis  of  actual  coincidence  of  spectra,  symmetrically. 

The  height  or  breadth  of  the  longitudinal  interference  band,  d  in  figure  6, 
is  not  greater  than  1.5  to  2  times  the  distance  apart  of  the  sodium  lines  at 
right  angles  to  the  band.  From  this  the  angular  divergence  of  the  breadth 
of  the  band  may  be  found,  since  \  =  D  sin  6,  where  X  is  the  wave-length  of 
light,  D  the  grating  space,  and  6  the  angle  of  diffraction.  Hence  for  the  two 
sodium  lines 

A0  =  AX/Z)cos  6 


14  THE   INTERFEROMETRY   OF 

Since  D  =  3$iXio-6,  cos  d=.gS6,  and  AX  =  6Xicr8;  therefore  A0=  1.7X10-" 
radians.  Since  the  width  of  the  band  is  about  twice  this,  it  will  be  68  seconds 
of  arc,  or,  roughly,  about  a  minute  in  breadth.  Within  the  strip,  when  the 
fringes  are  horizontal,  I  counted  about  five  of  them,  so  that  their  distance 
apart  would  be  about  14  seconds  of  arc. 

It  appears,  therefore,  that  rays  of  a  given  color,  say  of  the  wave-length  at 
D,  which  leave  the  grating  at  a  given  point  and  at  an  angle  of  about  one 
minute  in  the  plane  of  the  D  line,  are  still  in  a  condition  to  interfere ;  whereas 
one  would  anticipate  that  only  those  rays  which  lie  in  the  common  longitudinal 
axis  of  rotation  of  the  two  coincident  spectra,  symmetrical  to  this,  should  be 
in  this  condition.  Such  interference  should  not  be  appreciable,  since  the 
white  rays  are  independent  and  apparently  come  from  two  different  points 
of  the  slit.  If  we  consider  the  angular  deviation  of  pencils  of  parallel  rays 
crossing  the  grating  to  be  equivalent  to  the  divergence  of  their  respective 
optical  axes  at  the  collimating  lens  (about  45  cm.  in  focal  distance),  the  dis- 
tance apart  of  two  points  of  the  slit,  the  rays  of  which  are  still  able  to  produce 
interference,  is 

*  =  45XA0  =  45X i. 7X10^  =  7.6Xio-3  cm. 

or  nearly  o.i  mm.  Hence  points  of  white  light  in  the  slit  about  o.i  mm.  apart 
along  its  length  are  included  in  the  band  of  interferences  in  question,  extending 
in  colored  light  from  red  to  violet.  This  seemingly  anomalous  result  will  be 
fully  interpreted  at  another  opportunity. 

5.  Interference  of  the  corresponding  first-order  spectra  of  the  grating,  in 
the  absence  of  rotation. — This  apparatus  seemed  to  be  of  special  interest, 
since  the  rays  used  do  not  retrace  their  path  and  are  thus  available  for  experi- 


ments in  which  rays  traveling  in  one  direction  only,  are  needed.*  I  have 
tried  both  the  adjustments  given  in  figures  7  and  8,  the  latter,  since  the  rays 
are  more  nearly  normally  reflected  at  the  mirrors  M  and  N,  having  some 
advantages ;  but  the  other  succeeds  nearly  as  well.  The  difficulty  encountered 
is  a  curious  one  of  adjustment,  which  was  not  anticipated.  In  other  words, 
if  the  longitudinal  axes  of  two  identical  spectra  are  in  coincidence,  the  Fraun- 
hofer  lines  are  likely  to  be  at  a  small  angle  to  each  other  and  complete  inter- 

*  Cf.  Am.  Journal  of  Sci.,  xxxiv,  p.  101,  1912,  on  the  interferometry  of  an  air  column 
carrying  electrical  current. 


REVERSED  AND   NON-REVERSED    SPECTRA.  15 

ference  is  therefore  impossible.  Again,  if  the  spectrum  lines  are  in  coincidence, 
the  longitudinal  axes  usually  diverge  by  a  small  angle.  Furthermore,  the 
interferences  are  almost  always  eccentric  and  the  lines  hair-like,  indicating 
distant  centers.  I  have  not  succeeded  in  making  a  perfect  adjustment, 
systematically;  but  the  discrepancies  indicated  are  themselves  interesting  in 
their  bearing  on  the  subject  of  this  paper. 

In  figures  7  and  8,  L  is  a  vertical  blade  of  white  light  from  a  collimator 
with  fine  slit,  and  G  is  the  grating.  The  two  first-order  spectra  leaving  the 
ruled  face  at  the  line  g  strike  the  opaque  mirrors  M  and  N,  the  former  on  a 
micrometer  moving  the  mirror  parallel  to  itself.  From  M  and  N  the  rays 
reach  the  half-silvered  plate  of  glass  HS,  where  one  is  transmitted  and  the 
other  reflected  into  the  telescope  T.  The  coincident  rays  R  are  superfluous. 

After  placing  the  parts  and  roughly  adjusting  them  for  symmetry  with 
sunlight,  the  finer  adjustment  may  be  undertaken.  It  may  be  noticed  that 
the  two  systems  M  and  N,  and  G  as  well  as  HS,  can  be  used  for  further 
adjustment  separately.  All  are  provided  with  adjustment  screws  relatively 
to  rectangular  axes.  To  put  the  mirrors  M  and  N  in  parallel  and  in  the 
vertical  plane  with  the  grating  G,  the  half -silvered  plate  should  be  removed 


10 


and  replaced  by  a  small  white  vertical  screen  of  cardboard,  placed  at  right 
angles  to  the  direction  of  HS  in  figure  7  and  receiving  both  spectra.  A  fine 
wire  is  drawn  across  the  slit  to  locate  the  longitudinal  axis,  and  an  extra 
lens  may  be  added  to  the  collimator  and  properly  spaced  until  the  doublet 
insures  sharp  focussing.  Both  mirrors,  M  and  N,  are  now  rotated  on  hori- 
zontal axes,  until  the  longitudinal  black  lines  in  their  spectra  cease  to  diverge 
and  coincide  accurately.  G,  M,  N,  may  now  be  considered  in  adjustment. 
On  returning  the  half -silvered  plate,  HS,  it  in  turn  is  to  be  carefully  rotated 
around  horizontal  and  vertical  axes,  until  the  horizontal  black  line  in  the 
spectrum  and  the  sodium  line  (always  incidentally  present  in  the  arc  lamp) 
both  coincide.  But,  as  a  rule,  it  will  be  found  that  if  the  longitudinal  axes, 
ww,  figure  9,  coincide,  the  D  lines  cross  each  other  at  a  small  angle,  exagger- 
ated in  the  figure.  The  interferences,  when  found  by  moving  the  micrometer 
at  M,  are  usually  coarse,  irregular  lines,  indicating  a  center  not  very  distant 
and  located  on  the  level  of  a  band  where  the  D  lines  cross. 

On  the  other  hand,  if  the  D  lines  are  brought  to  coincidence  by  moving 
the  adjustment  screws  on  M  and  N  (which  throws  them  out  of  parallel),  the 
longitudinal  axes  ww,  w'w' ,  figure  10,  diverge  at  a  small  angle  and  the  inter- 
ferences are  found  in  a  vertical  band  where  the  lines  ww  and  w'w'  cross.  This 
band  is  relatively  wide,  however,  as  compared  with  the  cases  in  paragraphs  2 
and  3.  Nevertheless,  I  have  looked  upon  these  results  as  additional  proof 


16  THE   INTERFEROMETRY   OF 

of  the  possibility  of  interference;  for  in  neither  case  ought  they  to  occur  if 
the  spectra  are  not  quite  coincident  horizontally  and  vertically.  If  they  do 
occur,  it  would  at  first  sight  seem  that  a  certain  small  latitude  of  wave-length 
adjustment  is  permitted  even  with  light-waves. 

I  was  at  first  inclined  to  refer  the  cause  of  this  lack  of  simultaneous  parallel- 
ism to  the  grating  itself,  as  it  occurred  with  an  Ames  grating  ruled  on  glass, 
with  a  Michelson  reflecting  grating,  and  with  a  film  grating,  in  about  the 
same  measure.  But  subsequently,  on  adopting  the  method  of  figure  8,  the 
divergence  was  largely  removed  and  the  interferences  were  now  visible 
throughout  the  whole  of  the  spectrum.  The  discrepancy  is  probably  due  to 
insufficient  normality  of  the  plate  of  the  grating  to  the  incident  white  ray, 
since  one  of  the  rays  is  twice  reflected.  In  any  case  the  adjustment  of  the 
coincident  sodium  lines  must  be  very  accurate  if  the  fringes  are  to  be  sharp ; 
certainly  as  little  as  half  their  distance  apart  will  obscure  the  phenomenon. 

Though  the  spectra  are  bright,  the  interferences  are  not  as  good  as  with 
the  usual  method  (paragraph  i);  i.e.,  the  dark  lines  are  not  black.  Neither 
have  I  found  an  available  or  systematic  method  for  centering  the  fringes,  so 
that  the  lines  obtained  are  usually  delicate.  Again,  the  position  of  the  colli- 
mator,  both  as  regards  slit  and  lens,  is  here  of  very  serious  importance.  Any 
micrometeric  horizontal  motion  of  either,  in  its  own  plane,  will  throw  the 
fringes  out.  Finally,  the  whole  spectrum  travels  with  the  motion  of  the 
micrometer  mirror  M.  The  apparatus  is  thus  too  difficult  to  adjust  for  use, 
to  be  of  practical  interest  when  simpler  methods  are  at  hand.  The  effect  of 
tremors  acting  prejudicially  on  so  many  parts  is  exaggerated. 

6.  Conclusion. — The  phenomena  of  paragraphs  2,3,  and  4,  showing  definite 
and  characteristic  interference  in  case  of  two  coincident  spectra  crossed  either 
on  a  longitudinal  or  transverse  axis,  represent  the  chief  import  of  the  present 
chapter.  These  results  can  not  be  directly  due  to  the  diffraction  of  a  slit 
(regarding  the  line  of  coincidence  as  such),  owing  to  their  relatively  small 
magnitudes  and  their  independence  of  the  breadth  of  the  slit.  Since  there 
is  in  each  case  but  a  single  line  of  points  or  axis,  the  disturbance  of  which 
comes  from  identical  sources,  we  might  regard  the  image  of  this  line  in  the 
telescope  to  be  modified  by  the  diffraction  of  its  objective.  But  if  the  inter- 
ferences originated  in  this  way,  the  Fraunhofer  lines  of  the  spectrum  should 
show  similar  characteristics  and  the  diffraction  pattern  should  differ  from 
those  observed.  Thus  the  conclusion  is  apparently  justified  that  distinct  and 
independent  points  of  the  narrow  slit  whose  distance  apart  on  its  length  is 
not  greater  than  o.i  mm.  contribute  rays  to  the  field  of  interference  in  each 
of  the  colors  of  the  spectrum  (longitudinal  axes  coinciding). 

The  phenomenon  of  inversion  is  virtually  one  of  homogeneous  light,  the 
same  type  of  interference  occurring  in  each  color  from  red  to  violet.  When 
the  fringes  are  horizontal,  homogeneous  light  and  a  correspondingly  broad 
slit  would  replace  the  spectrum.  They  belong,  moreover,  to  the  elliptic 
category,  being  of  the  same  nature,  apart  from  their  limitations,  as  those 


REVERSED   AND   NON-REVERSED   SPECTRA.  17 

used  in  displacement  interferometry.  With  the  exception  of  the  points  lying 
on  the  longitudinal  axis  of  rotation  or  of  coincidence,  all  the  pairs  of  points 
of  the  two  coincident  spectra  owe  the  major  part  of  their  light  to  different 
sources;  i.e.,  the  points  of  the  superposed  spectra  are  not  colored  images  of 
one  and  the  same  point  in  the  slit. 

Again,  in  case  of  rotation  of  one  of  the  coincident  spectra  around  a  trans- 
verse axis  (Fraunhofer  line),  colors  which  differ  in  wave-length  by  about  half 
the  distance  apart  of  the  two  sodium  lines  seem  also  to  admit  of  interference. 
This  permissible  difference  of  wave-length  is  thus  relatively  about 

' 

or  less  than  o.i  per  cent.  The  character  of  these  interferences  is  distinctive. 
They  are  not  of  the  regular  elliptic  type,  but  arise  and  vanish  in  a  succession 
of  nearly  vertical  (parallel  to  slit),  regularly  broken  lines.  Later  observation, 
however.,  revealed  as  their  true  form  a  succession  of  long  spindles  or  needle- 
shaped  designs.  The  chief  peculiarity  observed  is  their  almost  scintillating 
mobility,  which  in  the  above  text  has  been  referred  to  the  inevitable  tremors 
of  the  laboratory.  It  is,  however,  interesting  to  inquire  into  the  conditions 
of  the  possibility  of  observable  beating  light-waves.  For  two  waves,  very 
close  together,  of  frequency  n  and  n'  and  wave-lengths  X  and  X',  if  V  is  the 
velocity  of  light,  the  number  of  beats  per  second  would  be 


Therefore  in  case  of  the  two  sodium  lines,  for  instance, 

w'-w  =  3Xiol°X6Xio-8/348oXio-12=5Xio11 

i.e.,  about  5  X  io10  beats  per  o.i  second,  the  physiological  interval  of  nickering. 
Naturally  this  seems  to  be  out  of  all  question,  even  if  one  is  confronting  a 
source  which  is  an  approach  to  a  mathematical  line.  The  endeavor  will  have  to 
be  made  to  produce  these  interferences  under  absolutely  quiet  surroundings. 
Their  appearance  is  altogether  singular  and  not  like  the  case  of  paragraph  4, 
where  there  is  also  perceptible  tremor,  or  with  the  general  case  of  trembling 
interference  patches,  with  which  I  am,  unfortunately,  all  too  familiar. 

In  this  place,  however,  it  is  my  sole  purpose  to  present,  at  its  face  value, 
an  observation  which  is  spatial,  independent  of  time  consideration;  and  the 
laterally  cramped  character  of  the  new  interference,  with  its  long,  hair-like 
lines  thrust  into  a  strip  less  than  half  the  distance  apart  of  the  sodium  lines, 
is  the  only  evidence  submitted.  If  the  coincident  path  of  two  rays  of  slightly 
different  wave-lengths,  X  and  X',  which  interfere,  is  x,  then  there  are  x/\  and 
x/\',  complete  waves  in  the  given  path,  and,  in  case  of  original  identity  in 
phase,  instantaneous  reenforcement  will  occur  when 


In  other  words,  at  the  wth  reenforcement 


18  REVERSED  AND   NON-REVERSED   SPECTRA. 

Hence,  since  X2  is  very  small  and  x  relatively  very  large,  the  small  value  of 
AX  (i.e.,  the  very  thin  strip  of  spectrum  within  which  the  phenomenon  occurs) 
is  apparent.  In  the  above  experiments  the  estimates,  in  round  numbers, 
were  AX  =  2.4Xicr8,  X2  =  36Xicr10.  Hence  if  n=i, 


so  that  one  reenforcement  would  have  to  occur  about  at  each  1.5  mm.  along 
the  rays.  Nevertheless,  the  formidable  difficulty  remains  to  be  investigated, 
viz,  why  these  nominally  beating  wave-trains,  with  an  infinitesimal  group 
period  (io~u  sec.),  could  be  recognized  at  all. 

The  characteristic  feature  of  the  new  phenomenon  is  this,  that  apart  from 
intensity  it  persists,  without  variation,  through  a  path-difference  of  over  5  milli- 
meters; i.e.,  through  15,000  or  20,000  wave-lengths.  It  follows,  since  the 
optical  paths  grating-mirror-grating  are  alone  significant,  that  two  individual 
light-waves  of  the  same  ray  over  15,000  wave-lengths  apart  are  still  appre- 
ciably identical.  Beyond  that  the  waves  under  consideration  no  longer 
correspond  in  orientation  and  can  not  interfere  in  a  way  to  produce  alterna- 
tions of  accentuated  brightness  and  darkness. 


CHAPTER  II. 


FURTHER  STUDY  OF  THE  INTERFERENCE  OF  REVERSED  SPECTRA. 

7.  Apparatus  with  one  grating. — The  different  methods  suggested  in  para- 
graph 3  were  each  tried  in  succession,  but  none  of  them  were  found  equally 
convenient  or  efficient  in  comparison  with  the  method  finally  used  in  the 
preceding  paper.  To  begin  with  the  annoyances  encountered  in  the  use  of  a 
reflecting  grating,  it  was  found  that  the  impinging  light  from  the  collimator 
and  the  reflected  doubly  diffracted  beam  from  the  grating  lie  too  close  together, 
even  if  all  precautions  are  taken,  to  make  this  method  of  practical  value.  The 
use  of  Rowland's  concave  grating  without  a  collimator  is  out  of  the  question, 
since  the  spectra  formed  on  the  circular  locus  of  condensation,  if  reflected 
back,  will  again  converge  into  a  white  image  of  the  slit,  colored  if  part  of  the 
spectrum  is  reflected.  The  plane-reflecting  grating,  though  not  subject  to 
this  law,  requires  a  collimator,  and,  since  marked  obliquity  of  rays  is  excluded, 
it  will  hardly  be  probable  that  the  elusive  phenomena  can  be  obtained  in  this 
way.  A  compromise  method,  in  which  both  the  reflecting  and  the  transmitting 
grating  are  used,  will  be  described  in  paragraph  10.  Though  apparently  the 
best  adapted  of  all  the  methods  used,  it  has  only  after  difficult  and  prolonged 
research  led  to  results.  These,  however,  proved  very  fruitful  in  their  bearing 
on  the  phenomena. 

For  first-order  spectra,  where  there  is  abundance  of  light  (it  is  often  difficult 
to  exclude  all  the  whitish  glare  in  the  field  of  the  telescope  completely),  the 
method  of  figure  1 1 ,  which  shows  normal  rays  only,  is  still  preferable.  Here 
the  impinging  collimated  beam  L  passes  below  the  opaque  mirror  m  and 
through  the  lower  half  of  the  grating  G.  The  diffracted  pencil  is  reflected 
nearly  normally  but  slightly  upward,  by  the  mirrors  M  and  N  (the  former 
carried  on  a  micrometer  slide) ,  to  be  again  diffracted  at  the  grating  and  there- 
fore to  impinge  as  definitely  colored  light  on  the  lower  edge  of  the  concave 
mirror  m  (about  1.5  to  2  meters  in  focal  distance),  whence  it  is  brought  to  a 
focus  at  F  and  viewed  by  the  strong  eyepiece  E.  Considerable  dispersion 
and  magnification  is  obtained  in  this  way;  indeed,  the  two  D  lines  stand  far 
apart  and  the  nickel  line  is  distinctly  visible  between  them.  There  must  be 
a  fine  hair  wire  across  the  slit  so  that  the  longitudinal  axes  of  the  spectra  may 
be  accurately  adjusted.  The  mirror  m  above  the  impinging  beam  must  be 
capable  of  rotation  about  a  vertical  and  a  horizontal  axis  in  order  that  the 
focus  F  may  be  appropriately  placed  between  M  and  N.  With  G  at  i  meter 
and  m  at  2  meters  from  F,  the  disposition  is  good.  The  micrometer  M  is 
easily  at  hand.  Though  the  direct  beam  may  be  screened  off,  the  glare 
reflected  back  from  the  grating  and  the  glare  from  the  objective  of  the  colli- 
mator are  not  excluded,  as  stated.  In  fact,  it  was  eventually  found  necessary 

19 


20  THE   INTERFEROMETRY  OF 

to  cany  this  pencil  in  an  opaque  tube  reaching  from  the  objective  of  the 
collimator,  as  far  as  the  grating. 

With  first-order  spectra  this  method  always  succeeded  satisfactorily,  and 
in  case  of  a  ruled  grating  the  phenomenon  is  exhibited  brilliantly,  if  the  paths 
CM  and  GN  are  optically  nearly  equal.  After  some  experience  it  is  fairly 
easy  to  find  it.  I  have  not,  however,  been  able  to  obtain  it  with  a  film  grating, 
even  after  using  a  variety  of  excellent  samples.  This  is  not  remarkable,  for 
the  film  grating  is  hardly  sufficiently  plane  to  produce  clear  regular  reflection, 
and  the  corresponding  paths  GM  and  GN  would  not,  therefore,  be  definite. 

Second-order  spectra  are  too  faint  and  can  not  be  seen,  unless  the  glare  is 
excluded  in  the  manner  stated.  All  modifications  of  the  method  seemed  with- 
out avail,  until  finally  the  light  was  led  from  the  collimator  objective  C, 
figure  n,  to  the  grating  G,  in  a  cylindrical  tube,  whereupon  both  the  glare 
from  the  objective  and  the  rearward  reflection  from  the  grating  were  effec- 
tively screened  off.  This  tube  must,  of  course,  lie  below  the  returning  pencil, 
i.e.,  it  must  not  (in  section)  cover  more  than  the  lower  half  of  the  grating. 
In  this  case  the  second-order  spectra,  though  faint,  were  seen  clearly;  but 


the  scintillating  interferences  could  not  be  observed  until  the  very  weak 
eyepiece,  E,  was  used  with  the  concave  mirror  m;  or  a  weak  telescope  with 
a  plane  mirror.  It  was  then  detected,  but  showed  no  essential  difference  from 
the  case  of  first-order  spectra.  The  larger  dispersion,  in  other  words,  was 
unavailable.  The  phenomenon  was  seen  most  distinctly  by  drawing  out  the 
eyepiece  of  the  telescope,  as  the  light  is  thereby  concentrated,  although  the 
Fraunhofer  lines  vanish.  Second-order  spectra  are  therefore  not  necessarily 
advantageous.  The  phenomenon  is  very  hard  to  find,  and  the  experiments 
were  persisted  in  only  to  obtain  the  result  under  different  conditions. 

The  tube-like  light  conductor  referred  to  above  is,  of  course,  advantageous 
in  case  of  first-order  spectra.  If  the  concave  mirror  is  used,  the  phenomenon 
may  even  be  seen  brilliantly  with  the  naked  eye. 

An  alternative  method  of  half-silvering  the  ruled  face  of  the  grating  and 
then  using  it  as  a  reflector  was  tried  with  success.  The  beam  of  parallel 
rays  from  the  collimator  L,  figure  12,  is  transmitted  by  the  grating  (ruled, 
half-silvered  face,  g  toward  the  mirrors  M  and  N)  and  the  two  diffracted 
beams  then  returned  by  the  opaque  mirrors  M  and  N,  to  be  in  turn  diffracted 
by  reflection  into  the  telescope  T.  In  fact,  this  method  succeeds  with  the 
unsilvered  grating;  for  the  rays  diffracted,  by  reflection,  from  the  ruled  face 
(toward  the  telescope),  but  not  very  well.  The  reflection  from  the  rear  face 


REVERSED  AND  NON-REVERSED  SPECTRA.         21 

of  the  grating  is  so  cut  up  by  the  strong,  stationary  interferences  that  it  is 
unavailable.  The  grating  plate  must,  of  course,  be  slightly  wedge-shaped, 
otherwise  all  the  spectra  would  be  superposed.  In  case  the  ruled  face  is 
half -silvered,  however,  the  stationary  interferences  are  practically  absent, 
while  two  strong  spectra  are  reflected  from  the  silvered  side.  The  phenome- 
non may  then  be  produced  at  all  distances  of  G  from  M  and  N  (2  meters  and 
less),  but  best  at  distances  within  i  meter.  It  is,  however,  frequently 
hard  to  find  unless  different  distances  apart  of  the  mean  D  lines  are  tested. 
This  may  be  due  to  the  fact  that  the  silver  film  is  not  quite  equally  thick. 

Besides  the  symmetrical  position,  gT,  figure  12,  the  two  corresponding 
unsymmetrical  positions  g'T'  were  tested  with  success;  and  it  appeared  that 
while  in  the  case  gT  the  phenomenon  is  virtually  linear,  dark  or  bright,  like 
a  Fraunhofer  line,  a  succession  of  dark  lines  inclined  to  the  vertical  may 
appear  for  the  unsymmetrical  position  g' T'.  Dark  lines  are  apt  to  be  broadened. 

Questions  relative  to  the  effect  of  oblique  incidence  were  also  tested  by 
aid  of  the  concave-mirror  method  shown  in  figure  n,  the  white  light  from  C 
to  G  being  conducted  in  an  inch  tube  of  pasteboard,  immediately  under  the 
concave  mirror,  m.  Figure  13,0,  shows  the  general  disposition  of  apparatus. 


\3 


The  angle  of  incidence  *  is  gradually  increased,  until  the  return  rays  from  N 
meet  the  grating  at  nearly  grazing  incidence.  No  essential  difference  in  the 
phenomenon  was  observed,  however,  except  that  it  was  apt  to  be  broader 
in  the  non-symmetrical  positions  and  to  suggest  fine  new  lines  in  parallel 
with  the  old.  In  a  return  to  the  symmetrical  position,  sharp  lines  were 
especially  distinct,  usually  showing  one  dark  and  two  bright  lines,  while  two 
dark  and  one  bright  occurred  less  frequently.  It  could  be  seen  quite  vividly 
with  the  naked  eye.  When  the  telescope  was  used  and  the  ocular  drawn 
far  forward,  the  multilinear  form  was  often  suggested.  On  broadening  the 
slit  the  black  lines  vanish  first  and  a  flickering  band  remains  after  the  Fraun- 
hofer lines  are  gone.  Finally,  the  phenomenon  could  be  seen  even  when  the 
longitudinal  axes  of  the  spectra  were  not  quite  coincident,  but  it  rapidly 
became  fainter  in  intensity. 

Figure  13,  b,  suggests  a  method  of  using  a  reflecting  grating,  either  plane 
or  (possibly,  if  the  incident  light  is  parallel)  concave,  for  the  production  of 
the  phenomenon.  G  is  the  grating,  receiving  the  collimated  white  light,  L, 
which  is  diffracted  toward  M  and  N,  thence  reflected  (at  a  different  elevation) 
back  to  G,  to  be  again  diffracted  towards  T,  above  or  below  the  direct  beam, 
where  it  is  observed.  I  have  not,  however,  been  able  to  obtain  results  with 
these  methods  owing  to  subsidiary  difficulties. 


22  THE   INTERFEROMETRY   OF 

8.  Observations  and  experiments  with  a  single  grating.  —  On  considering 
figure  ii,  it  will  be  seen  that  the  doubly  reflected,  doubly  diffracted  rays  are 
also  in  a  condition  to  interfere.  Thus  the  rays  GMGNG  and  GNGMG  have 
identical  path-length,  or  at  least  path-difference;  but  it  is  improbable  that 
superimposed  on  the  strong  spectra  this  effect  could  be  seen,  for  the  reflec- 
tion from  the  ruled  face  of  the  grating  is  very  slight  and  the  divergent 
spectra  have  weakened  seriously.  The  scintillating  interferences,  on  the 
other  hand,  are  much  brighter  than  the  superposed  spectra.  Such  interfer- 
ences, also,  should  be  independent  of  the  play  of  the  micrometer  M,  since 
the  path-difference  of  these  beams  is  not  changed  thereby,  each  being  identi- 
cally lengthened  or  shortened.  Furthermore,  the  interposition  of  a  thick 
plate-glass  compensator  in  CM  should  have  no  effect.  Neither  of  these  infer- 
ences applies  for  the  phenomenon  in  question,  which  persists  for  a  definite 
displacement  of  M,  only,  and  the  introduction  of  a  compensator  requires  the 
usual  equivalent  displacement  of  M,  within  the  range  of  the  phenomenon. 
Finally,  the  interferences  relatively  to  a  phenomenon  produced  by  double 
diffraction  would  not  be  modified. 

Many  experiments  were  made  to  ascertain  the  path-difference  within  which 
the  phenomenon  is  visible.  This  can  not  be  accurately  determined,  since  it 
is  a  question  of  stating  when  an  observation,  which  is  becoming  rapidly  less 
distinct,  has  actually  vanished.  Moreover,  any  imperfection  of  the  microm- 
eter throws  out  the  coincidence  of  longitudinal  spectrum  axes,  while  a  read- 
justment breaks  the  continuity  of  the  micrometer  displacement,  or  reading. 
Results  were  obtained  as  follows,  for  example,  AN"  being  the  displacement  of 
the  mirror  M: 

With  telescope  ............................  AAr  =  0.34,  0.45,  0.41  cm. 

With  concave  mirror  and  lens  ..............  0.45,  0.35,  0.41  cm. 

With  concave  mirror  and  adjustment  .......  0.50      to     0.60  cm. 

The  low  readings  are  due  to  the  rnicrometric  wabbling  of  the  micrometer 
slide.  Since  AAT  is  the  double  path-difference,  the  number  of  wave-lengths 
in  question  may  be  put 


i.e.,  the  distances  along  the  ray  are  15,000  to  20,000  wave-lengths  apart, 
about  as  estimated  in  the  above  paper.  This  is  the  characteristic  feature  of 
the  phenomenon. 

Between  its  extreme  ranges  of  visibility  the  appearance  of  the  phenomenon 
scarcely  changes.  It  ceases  to  be  visible  rather  suddenly;  and  this  is  to  be 
expected,  since  we  are  dealing  directly  with  two  wave-trains  displaced  rela- 
tively to  each  other.  It  is  visible  for  a  wide  slit  even  after  the  Fraunhofer 
lines  vanish.  It  disappears  by  decreasing  in  width,  when  the  slit  is  closed. 
If  the  ocular  of  the  telescope  is  drawn  out,  the  phenomenon  may  even  be 
observed  after  the  Fraunhofer  lines  have  vanished,  in  the  dark,  stringy  spec- 
trum of  an  extremely  fine  slit.  When  the  longitudinal  axis  of  the  spectrum 
is  indicated  by  a  fine  wire  across  the  slit,  the  adjustment  consists  in  bringing 


REVERSED   AND   NON-REVERSED   SPECTRA.  23 

the  black  longitudinal  lines  of  the  two  spectra  together.  The  question  thus 
arises  how  close  this  coincidence  is  to  be.  When  the  phenomenon  is  sharp, 
it  has  been  found  possible  to  displace  the  two  black  lines  so  that  a  fine, 
bright  strip  of  spectrum  may  just  be  seen  between,  without  quite  destroying 
the  interferences.  Naturally  they  are  then  much  weaker.  This  result  is  in 
harmony  with  the  observations  made  on  rotating  one  spectrum,  on  a  longi- 
tudinal axis,  1 80°  with  reference  to  the  other. 

Since  the  phenomenon  was  originally  produced  with  sunlight,  it  might  be 
supposed  that  the  edges  of  the  Fraunhofer  line,  under  conditions  of  tremor, 
would  interfere  with  each  other  as  indicated  in  figure  14,  where  A  is  one  and 
B  the  other  of  the  two  superposed  spectra.  The  change  of  wave-length  is 
suggested  by  the  slant  of  lines  on  the  ~  / 

diagram.    In  such  a  case,  whereas  the    ^  r- . y  £-   _^:_   v  {L — , ,_.     i) 

conditions  a  and  c  would  show  bright  $  ^o~°  f====^r.  <Q  ~1"'1 — r  v  '~i  ""  T 
overlapping  spectra,  the  dark  line  would  OT  dk  or 

appear  under  condition  b.     But  even 

in  this  case,  lines  of  slightly  different  wave-length  would  have  to  interfere 
with  each  other.  The  crucial  test  was  made  by  using  an  arc-lamp  spectrum, 
and  it  was  then  found  that  the  phenomenon  appeared  as  well  as  with  sunlight. 

A  further  question  at  issue  is  the  breadth  of  spectrum  needed  to  produce 
the  phenomenon;  for  the  observed  breadth  would  be  influenced  by  the  quiver 
of  the  apparatus.  With  this  end  in  view,  different  lines  of  the  spectrum  were 
placed  in  full  coincidence,  and  it  was  found  that  for  none  of  the  secondary 
lines  in  the  orange-yellow  spectrum  was  it  extinguished  or  even  modified. 
If,  however,  the  corresponding  D  lines  of  the  spectra  (DiZY;  D2  D2f)  were 
superposed,  the  phenomenon  in  these  experiments  played  like  a  wavy  strip  at 
their  edges  only.  Sometimes  a  bright  line  flashed  through  the  middle  of  the 
coincident  lines.  One  would  conclude,  therefore,  that  the  part  of  the  spectrum 
used  in  producing  these  interferences  is  not  much  broader  than  either  the 
DI  or  DZ  lines,  while  the  other  marked  lines  in  the  orange-yellow  are  too 
narrow  to  appreciably  influence  it.  These  results  will  be  greatly  amplified 
in  the  work  done  with  two  gratings  below. 

A  corresponding  experiment  was  now  made  with  sodium  light.  To  obtain 
a  sufficiently  intense  source,  solid  caustic  soda  was  volatilized  between  the 
carbons  of  the  electric  arc,  A  and  B,  figure  12,  or  the  corresponding  case  in 
figure  1 1 .  On  drawing  the  carbons  apart,  strong  D  lines  were  seen,  in  the  entire 
absence  of  an  arc  spectrum,  at  first  so  broad  as  to  be  self-reversing.  Gradu- 
ally they  became  finer  and  eventually  reached  the  normal  appearance  of  the 
DI,  Dz  lines.  In  order  to  facilitate  adjustment  and  with  the  object  of  obtain- 
ing cases  correlative  with  the  results  for  the  dark-line  spectrum,  a  beam  of 
sunlight  (as  at  L,  figure  12)  was  introduced  between  the  carbons  and  the  phe- 
nomenon established  faultlessly  in  the  usual  way.  The  pencil  of  sunlight  was 
then  screened  off  and  the  arc  light  substituted,  or  the  two  were  used  together. 

These  observations  seemed  to  show  that  when  the  normal  DI  or  D2  lines 
were  placed  in  coincidence,  the  thread-like  phenomenon  fails  to  appear  with 


24  THE   INTERFEROMETRY  OF 

all  the  characteristics  visible  in  the  case  of  sunlight.  When  the  slit  is  broad- 
ened an  alternation  of  brightness,  or  flicker  of  light,  may  be  detected  vaguely. 
With  a  slit  of  proper  width  to  show  the  Fraunhofer  lines  all  this  seemed  to 
vanish.  The  actual  phenomenon  was  therefore  apparently  not  reproduced 
or  improved  either  by  homogeneous  light  or  by  widening  the  slit.  Such  exper- 
iments alternating  with  sunlight  were  made  at  considerable  length,  but  the 
adaptation  of  methods  for  two  gratings  discussed  in  paragraph  io  will  never- 
theless throw  out  this  conclusion. 

If  the  narrow  sodium  line  is  broadened  by  adding  fresh  sodium  at  the  car- 
bon, so  that  the  yellow  spectrum  is  again  self -reversed,  the  phenomenon  plays 
with  extreme  vividness  around  either  of  the  reversed  and  coincident  Di  or 
Dz  lines,  or  even  within  the  black  line  in  question,  if  narrow.  But  here  the 
light  is  no  longer  homogeneous.  Sometimes  when  the  solar  spectrum  is  used, 
a  black  line  preponderates;  in  other  adjustments  a  flashing  bright  line  is  in 
place;  but  the  reason  for  this  can  not  be  detected  by  the  present  method. 

9.  Inferences. — If  the  wave-length  of  the  two  spectra  is  laid  off  in  terms 
of  the  angle  of  diffraction,  6,  measured  in  the  same  direction  in  both  cases, 
the  graph  will  show  two  loci  as  in  figure  15,  a,  intersecting  in  the  single  point 
of  coincident  wave-lengths  XQ.  It  appears,  however,  as  if  the  wave-lengths  at 
<f>i  and  w,  <f>z  and  ^4,  are  still  in  a  condition  to  interfere.  The  phases  <pi  and 
<pz,  <ps  and  <p4,  differ  because  of  path-difference  introduced  for  instance  at  the 
micrometer,  the  phases  <p\  <ps,  tpz  <p*  differ  because  of  color  differences,  having 
passed  through  refracting  media  of  glass  and  air.  Probably  the  phase-differ- 
ence <f>i—<f>3=<p2— (?4,  these  having  the  same  color-difference;  and  <pi—<f>z= 
<pi—<f>i,  having  the  same  path-difference.  At  Xo,  0o,  the  two  phases  <?0  are 
due  to  path-difference  only. 

To  allude  again  to  the  question  of  beats:  if  ten  beats  per  second  are  dis- 
cernible, the  beating  wave-trains  in  the  case  of  the  given  grating  would  be 
only  6Xio-10  second  of  arc  apart  in  the  spectrum.  If  the  phenomenon  has 
a  breadth  of  3XIO*8  cm.  in  wave-length,  as  observed,  then  the  number  of 
beats  in  question  will  be  2.5  X  io11  per  second.  All  this  is  out  of  the  question, 
so  far  as  the  phenomenon  appreciable  to  the  eye  is  concerned.  If  beats  were 
due  to  a  difference  of  -velocity  resulting  from  the  dispersion  of  air,  and  if  T 
is  the  period  of  the  beats,  X  the  mean  wave-length,  8-  the  difference  of  the 
reciprocal  indices  of  refraction,  we  may  write 

X 


If,  furthermore,  n  =  A-B/\*,  where  5=  1.34X10-",  5X  =  2.4X10-*, 

T  =     X4        _  1.3  X  io-17  _  _  6 

1     2vBd\     2X3Xio10Xi.34Xio-14X2.4Xio-*  ~>7XI° 


Ni=  i.4X  io6  beats  per  sec. 
which  would  also  be  inappreciable. 


REVERSED   AND   NON-REVERSED   SPECTRA.  25 

If  both  the  difference  of  wave-length  and  wave-velocity  are  considered, 
we  should  have  for  the  first  spectrum  v  and  n,  and  for  the  second  spectrum 
v  and  n'.  The  conditions  would  be  left  unchanged,  if  the  second  velocity  is 
taken  equal  to  the  first  and  the  frequency  n'(v'/v)  replaced  by  n'.  From  this 
it  follows  that  the  number  of  beats  N  is  nearly 


If  5X  is  considered  negative,  if  p.=A—  B/\2  and  the  multipliers  /*  and  n2  be 
neglected, 


which  is  the  difference  of  the  two  cases  above  computed.  As  the  first  is  very 
large  compared  with  the  second,  the  visibility  of  the  phenomenon  is  not 
changed. 

The  theory  of  group  waves  usually  introduces  a  factor  2.    Thus  if  Xi,  vi,  n\, 
be  the  group  wave-length,  velocity,  and  frequency, 


or, 

or  with  the  above  data 


results  otherwise  like  the  above  and  without  bearing  here.  There  is  a  possible 
question  whether  differences  of  wave-length  due  to  velocity  and  not  to  period 
can  be  treated  as  dispersion. 

The  occurrence  of  forced  vibrations  has  also  been  looked  to  as  an  explana- 
tion. Though  here  again,  even  if  the  spectra  are  almost  always  of  unequal 
intensity,  the  reason  for  the  preponderance  of  one  would  have  to  be  stated. 
True,  equal  mean  strength  is  not  equivalent  to  equal  instantaneous  strength. 
In  the  case  of  forced  vibrations,  however,  if  the  harmonic  forces  of  one  spec- 
trum are  F=A  cospt  (forced,  T=2ir/p),  of  the  other  F=A'cosqt,  (free, 
T=2ir/q)  and  there  is  no  friction,  the  resulting  harmonic  motion  will  be 
given  by 


Now  if  we  regard  the  case  of  figure  15,  on  one  side  of  the  line  of  coincidence 
Xo,  q2>p2',  on  the  other  side,  p2>q2.  Hence,  whenever  a  brilliant  line  flashes 
out  due  to  coincident  phases,  there  should  also  be  a  black  line  due  to  opposi- 
tion; and,  in  fact,  when  the  phenomenon  is  produced  under  conditions  of 
perfect  symmetry  of  the  component  beams,  this  seems  to  be  its  character; 
i.e.,  the  enhanced  line  cuts  vertically  across  the  breadth  of  the  spectrum. 
The  case  q2  =  p2,  being  of  infinitely  small  breadth,  would  not  be  visible.  It  is 
not  to  be  overlooked,  however,  that  in  certain  adjustments,  particularly  in 


26  THE    INTERFEROMETRY   OF 

the  non-symmetrical  case  of  figure  13,  more  than  two  black  lines  frequently 
occur.  (Cf .  §  1 5.)  These  accessory  lines  are  ordinarily  very  thin  and  crowded 
on  one  side  of  the  phenomenon  only.  It  is  thus  merely  the  prevalent  occur- 
rence of  paired  dark  and  bright  lines  that  are  here  brought  to  mind.  Again, 
the  suggestion  of  many  oblique  lines  has  occurred  in  some  of  the  observations. 
These  would  be  quite  unaccounted  for. 

Finally,  many  attempts  were  made  to  find  whether  the  phenomenon  would 
occur  again  beyond  its  normal  range  of  about  2X0.5  cm.  of  displacement. 
But,  though  the  micrometer  screw  actuating  the  mirror  M  was  effectively 
2X3  cm.  long,  no  recurrence  could  be  found.  At  the  ends  of  its  range  the 
phenomenon  drops  off  rather  abruptly. 

None  of  the  inferences  put  forward  adequately  account  for  the  phenome- 
non as  seen  with  a  single  grating,  as  a  whole.  In  this  dilemma  I  even  went 
so  far  as  to  suppose  that  a  new  property  of  light  might  be  in  evidence.  One 
feature,  it  is  true,  has  been  left  without  comment,  and  that  is  the  width  of 
the  slit-image.  If  ab,  figure  15  b,  is  the  angular  width  (d&)  of  this  image, 
the  case  of  figure  15  a  should  be  additionally  treated  in  terms  of  figure  156. 
But  within  the  limits  of  the  present  method 
of  experiment,  with  but  one  grating,  this 
circumstance  seems  to  offer  no  clue.  If,  for 
instance,  the  spectra  actually  coincide  in 
color  throughout  their  extent,  as  in  ordi- 
nary interferences,  the  interference  patterns 
should  be  enormous,  for  the  path-difference 
may  be  zero.  The  invariability  of  the  present  phenomena  as  to  size  within  its 
long  range  of  presence,  the  occurrence  of  intensely  sharp  and  bright  or  dark 
single  lines,  with  a  distance  (d&)  much  less  than  the  distance  apart  of  the  DI,  D% 
lines,  is  in  no  way  suggested  by  the  width  of  slit-image.  Moreover,  in  spite 
of  its  persistence,  the  interference  phenomenon  of  reversed  spectra  has  the 
sensitiveness  of  all  interferences.  Slight  tapping  on  the  massive  table  throws 
it  out  altogether.  Clearly,  therefore,  a  modification  of  method  is  essential  if 
new  light  is  to  be  thrown  on  the  phenomenon,  and  from  this  viewpoint  a 
separation  of  the  two  diffractions  seems  most  promising. 


10.  Apparatus  with  two  gratings.— All  the  varied  experiments  described  in 
the  preceding  paragraph  failed  to  show  any  essential  modification  of  the  linear 
interference  pattern  obtained.  In  a  measure  this  was  to  be  anticipated,  inas- 
much as  both  diffractions  take  place  at  the  same  grating.  It  therefore  seemed 
promising  to  modify  this  limitation  of  the  experiments,  although  the  difficulty 
of  finding  the  phenomena  would  obviously  be  greatly  increased.  The  separa- 
tion of  the  two  diffractions,  however,  seemed  to  be  alone  capable  of  resolving 
the  phenomenon  into  intelligible  parts. 

In  the  present  method  the  glass  grating  G,  figure  16,  receives  the  white 
beam  L  from  the  collimator,  which  is  then  diffracted  to  the  opaque  mirror  M 


REVERSED   AND   NON-REVERSED   SPECTRA. 


27 


(on  a  micrometer  slide)  and  N,  thence  to  be  reflected  to  the  reflecting  grating 
G',  plane  or  curved.  Here  the  two  beams  of  the  identically  colored  light 
selected  are  again  diffracted  to  the  telescope  or  lens  at  T.  Since  the  gratings 
G,  G',  rarely  have  the  same  grating  constant,  their  proper  position  must  be 
found  by  computation  and  trial.  In  my  work  the  distances  to  the  line  of 
mirrors  NM  were  165  cm.  for  G  and  90  cm.  for  G' .  This  method  automati- 
cally excludes  the  direct  beam  a  and  all  glare,  and  gives  excellent  spectra  both 
in  the  first  and  second  orders.  The  use  of  two  gratings,  however,  introduces 
the  difficulties  of  adjustment  specified,  as  the  two  D  doublets  corresponding 
to  AT  and  M  will  not,  as  a  rule,  be  parallel  and  normal  to  the  longitudinal 
axes  of  the  spectrum,  unless  all  cardinal  features,  like  the  rulings  and  their 
planes,  are  quite  parallel.  If  the  grating  is  not  normal  to  the  impinging 
beam,  the  axis  of  the  corresponding  spectrum  is  a  curved  line.  The  spectra 
are,  moreover,  likely  to  be  unequally  intense,  a  condition  not  infrequent 
even  in  the  preceding  method.  It  is  possible  that  this  may  be  due  to  the 
grating  itself,  but  probably  unequal  parts  of  the  corresponding  beams  are 


16 


used  in  the  two  cases,  or  the  mirrors  are  unequally  good.  As  a  result,  in  my 
earlier  work  I  was  not  able  to  produce  the  phenomena  with  two  gratings, 
after  many  trials,  in  spite  of  the  clearness  of  the  overlapping  spectra;  but 
the  same  serious  difficulties  are  encountered  whenever  interferences  are 
produced  from  two  independent  surfaces. 

Later,  having  added  a  number  of  improvements  to  facilitate  adjustments, 
I  returned  to  the  search  again  and  eventually  succeeded.  There  are  essen- 
tially four  operations  here  in  question,  supposing  the  grating  G  approximately 
in  adjustment.  By  aid  of  the  three  adjustment  screws  on  each  of  the  mirrors 
M  and  N,  figure  16,  the  fine  wire  drawn  across  the  slit  may  be  focussed  on  the 
grating,  if  an  extra  lens  is  added  to  the  collimator  and  the  black  horizontal 
shadows  of  that  wire,  across  the  corresponding  spectra,  placed  in  coincidence. 
The  grating  G'  is  then  to  be  moved  slowly  fore  and  aft,  normal  to  itself,  on 
the  slide,  so  that  the  position  in  which  the  sodium  lines  are  nearly  in  coinci- 
dence to  an  eye  placed  at  the  telescope,  T,  may  be  found.  The  grating  G' 
is  next  to  be  slowly  rotated  on  a  line  (parallel  to  LT)  normal  to  its  surface, 
to  the  effect  that  the  black  axes  of  both  spectra  (i.e.,  the  spectra  as  a  whole) 
may  coincide.  This  must  be  done  accurately,  and  the  last  small  adjustments 
may  be  made  at  the  screws  controlling  M  and  N.  Finally,  the  micrometer 


28 


THE   INTERFEROMETRY   OF 


a 


slide  carrying  M  is  to  be  moved  fore  and  aft  until  the  interferences  appear. 
These  operations  are  difficult  even  to  an  experienced  observer.  The  fringes 
are  very  susceptible  to  tremors,  and  only  under  quiet  surroundings  do  they 
appear  sharply.  At  other  times  they  move,  as  a  whole,  up  and  down  and 
intermittently  vanish. 

The  fringes  so  obtained,  figure  17,  were  totally  different  from  the  preceding 
and  consisted  of  short,  black,  equidistant,  nearly  horizontal  lines  across  the 
active  yellow  strip  of  spectrum,  at  the  axis  of  coincidence.  The  strip  was 
about  of  the  same  width  as  above.  Thus  the  pattern  presented  the  general 
appearance  of  a  barber's  pole  in  black  and  yellow,  the  width  being  less  than 
the  sodium  interval,  D\,  DZ,  and  the  distance  apart  of  fringes  usually  smaller. 
They  were  visually  in  motion  up  and  down,  rarely  quiet,  no  doubt  owing  to 
tremor.  Since  the  fringes  were  nearly  horizontal  or  less  than  30  degrees  in 
inclination,  it  was  possible  to  enlarge  the  width  of  the  slit  without  destroying 
them,  as  in  case  of  the  hair-like  vertical  fringes  in  paragraph  2  above.  In 
this  way  a  breadth  of  strip  greater  than  the  distance  Di,  DZ,  could  be  obtained 
with  sunlight  or  arc  light,  though  a  moderately  fine  slit  was  still  desirable. 

—  - '  _",    0< 


c- 


d 


17 


f 


h 


In  general,  the  characteristics  noted  above  were  again  observed.  Thus  on 
moving  the  micrometer  screw  controlling  M,  the  interferences  appeared  rather 
abruptly.  They  vanished  in  a  similar  manner,  after  about  0.4  cm.  or  more 
of  the  micrometer  screw  had  been  passed  over.  In  other  words,  the  fringes 
remain  identical  for  a  path-difference  of  about  2X0.4  cm.,  or  nearly  15,000 
wave-lengths. 

If  we  call  the  four  D  lines  available  in  the  two  solar  spectra  D\,  DZ,  D\,  D'z, 
respectively,  a  number  of  curious  results  were  obtained  on  placing  them 
variously  in  approximate  coincidence.  Thus  figure  17  a,  when  each  D  line 
of  one  spectrum  coincides  with  the  mate  of  the  other  (Dit  D'2;  D'i,  D2),  equi- 
distant dots,  surrounded  apparently  by  yellow  luminous  circles,  appeared 
between  the  two  doublets.  On  widening  the  slit  the  dots  changed  to  a  grating 
of  nearly  horizontal  lines  covering  the  strip  DI,  D2,  figure  176.  The  lines  in 
one  part  of  the  slit  seemed  to  slope  upward  and  in  another  to  slope  downward. 
With  a  large  telescope  the  phenomenon  was  more  dim  and  quiet,  apparently. 
The  fringes  often  lie  in  more  definite  focal  planes  and  cease  to  be  visible  when 
the  ocular  of  the  telescope  is  far  outward,  differing  from  the  case  above. 

The  phenomenon  of  chief  interest,  however,  was  observed  (figure  17  c)  in 
placing  two  identical  D  lines  in  coincidence  (Di;  D2  D'2;  D'i).  The  fringes 


REVERSED   AND   NON-REVERSED   SPECTRA.  29 

were  then  seen  across  the  coincident  lines,  now  no  longer  visible,  quite  inde- 
pendent of  the  absence  of  light.  This  would  seem  to  mean  that  the  otherwise 
quiet  ether  within  the  black  line  is  stimulated  into  vibration  by  the  identical 
harmonic  motions  of  the  bright  fields  at  and  beyond  the  edges  of  the  line 
(diffraction).  The  question  will  presently  be  broached  again  in  a  different 
way.  Here  I  may  note  that  in  the  above  cases  of  transverse  lines  (§  8)  it  is 
often  possible  to  observe  a  very  fine  parallel  yellow  line  within  the  coincident 
DZ,  D'z,  or  Di,  D'i,  doublets,  excited,  therefore,  in  the  dark  space  and  splitting 
the  line. 

The  experiments  were  now  repeated  with  the  sodium  arc,  and  these  also 
gave  some  striking  results.  Thus  in  the  case  of  figure  17  d  the  lines  were 
separated,  but  the  yellow  striations  seemed  to  show  across  the  dark  space 
between  DZ  and  D'2.  When  the  yellow  light  was  too  weak,  cross-hatchings 
were  seen  only  across  D'z,  as  in  figure  17  e.  Frequently  the  phenomenon 
figure  1 7  /  occurred  on  broadening  the  slit,  in  which  D2  and  D'z  interfered, 
but  only  D'z  was  marked.  Screening  off  D2  (left  mirror)  at  once  removed  the 
fringes.  I  have  interpreted  this  observation  as  the  result  of  parallax,  due  to 
the  fact  that  the  lines  and  the  interferences  are  seen  in  different  focal  planes. 


On  the  basis  of  these  results  one  might  with  some  plausibility  adduce  the 
following  remarks  in  explanation  of  the  phenomenon:  In  figure  18  a,  let  5i 
and  52  be  the  overlapping  reversed  spectra  and  let  the  line  of  symmetry  be 
at  Xi,  X2.  Then  if  identical  ether  vibrations  can  react  on  each  other  across  a 
narrow  ether  gap,  rays  as  far  as  X'i,  X'2  and  X"i,  X"2  being  of  identical  source 
and  wave-length,  respectively,  are  still  in  a  condition  to  interfere.  There 
would  then  be  three  groups  of  interferences,  Xi  X2,  X'i  X'2)  X%  X"2.  If,  figure 
1 8  b,  all  are  in  phase,  we  should  have  a  brilliant  line;  if  all  are  in  opposite 
phases,  a  dark  line  on  the  principle  of  figure  18  c.  Naturally,  if  wave-trains 
react  on  each  other  across  an  ether  gap,  small  as  compared  with  the  D\,  DZ 
interval,  the  assumption  made  above  relative  to  interference  of  different 
wave-lengths  is  superfluous.  My  misgiving  in  the  matter  arises  from  the 
misfortune  of  having  taken  down  the  original  apparatus,  for  modification, 
and  having  since  been  unable  to  reproduce  them  with  anything  like  the 
decisiveness  with  which  they  were  at  first  apparently  observed.  I  can  not 
now  be  certain  whether  what  occurred  was  actually  what  I  seemed  to  see, 
or  whether  the  broad  illumination  of  the  sodium  flash  (broad  individual 
lines,  DI  to  D2,  virtually  a  continuous  spectrum)  may  not  have  misled  me. 
The  experiments  were  continued,  as  follows. 


30 


THE   INTERFEROMETRY   OF 


11.  Experiments  continued.  New  interferometer. — At  the  outset  it  was 
necessary  to  ascertain  the  reason  for  the  difference  of  the  phenomena,  as 
obtained  with  one  grating  in  paragraph  8  and  with  two  gratings  in  para- 
graph 10.  As  the  probable  cause  is  a  lack  of  parallelism  of  the  rulings  in  the 
latter  case,  it  was  necessary  to  remount  the  second  grating  G'  in  the  manner 
shown  in  figure  19.  Here  A  A  is  a  baseboard,  capable  of  sliding  right  or  left 
and  of  rotating  on  a  horizontal  axis  parallel  to  the  grating.  The  latter  (in  a 
suitable  frame)  is  held  at  the  bottom  by  the  axle,  e,  normal  to  the  grating 
and  by  the  two  set-screws  a  and  b  carried  by  the  standards  c  and  d.  Thus 
the  grating  could  be  rotated  around  an  axis  normal  to  its  plane.  At  first  a 
Michelson  plane-reflecting  grating  G'  and  a  telescope  were  used,  as  in  figure 
16;  but  it  was  found  preferable  (fig.  20)  to  use  a  Rowland  concave  reflecting 
grating  G',  with  the  strong  lens  at  T,  the  grating  receiving  a  beam  of  parallel 
rays  of  light  for  each  color  from  the  collimator  and  first  grating  G.  In  this 
case,  with  sufficiently  high  dispersion,  a  large,  strong  field  was  obtained,  in 
which  even  the  very  fine  lines  of  the  solar  spectrum  were  quite  sharp.  Rotating 
grating  G'  around  a  parallel  horizontal  axis,  like  A  A,  figure  19,  made  little 


0 

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^     c4 

19 


difference,  relatively  speaking;  but  rotation  around  the  axis  e,  normal  to  its 
plane,  carried  out  by  actuating  a  and  6  in  opposite  directions,  made  funda- 
mental differences  in  the  appearance  of  the  phenomenon  and  eventually 
suggested  a  new  interferometer  for  homogeneous  light. 

The  adjustments  are  the  same  as  in  case  of  figure  i6,G  being  the  transparent 
grating,  except  that  G'  is  now  a  concave  grating  and  T  a  strong  eyepiece. 
The  distances  G'T  and  GT  were  of  the  order  of  i  and  2  meters. 

On  rotating  the  grating  G'  on  an  axis  normal  to  its  face,  from  a  position  of 
slight  inclination  of  the  rulings  toward  the  left,  through  the  vertical  position, 
to  slight  inclination  to  the  right,  the  fringes  passed  through  a  great  variety 
of  forms,  to  be  described  in  detail  in  §  13  below.  Difference  of  focal  planes 
between  the  Fraunhofer  lines  and  the  interferences  were  common,  so  that 
effects  of  parallax  were  apt  to  occur.  Thus  when  Dz  and  D'2  coincide,  the 
ladder-like  phenomenon  may  lie  between  D't  and  D\\  or  the  ladder  may  pass 
obliquely  between  the  Dz  Dl  and  D\  D'z  doublets.  The  first  experiment 
with  the  new  and  powerful  apparatus  (plane  transparent  grating  6",  grating 
space  3SiXio-6  cm.,  and  the  concave  reflecting  grating  G',  grating  space 
I73XIQ-6  cm.,  fig.  20)  was  made  with  the  object  of  verifying,  if  possible,  the 


REVERSED   AND   NON-REVERSED   SPECTRA.  31 

reaction  of  parallel  ether  wave-trains  on  each  other  across  a  very  narrow 
ether  gap.  The  sodium  arc  lamp  was  used  as  a  source  of  light.  The  results 
as  a  whole  were  negative,  or  at  least  conflicting.  Usually  when  strong  inter- 
ferences were  observed  for  coincident  positions  of  DZ,  D'%,  for  instance,  there 
was  no  passage  of  fringes  across  the  dark  space  when  DZ  and  D'%  were  slightly 
separated.  At  the  beginning  of  the  work  (possibly  as  the  result  of  lines 
broadened  by  a  flash  of  sodium  light)  the  stretch  of  interference  fringes  across 
the  dark  space  was  certain;  but  such  evidence  is  not  quite  trustworthy,  for 
a  continuous  spectrum  (i.e.,  lines  broadened  by  the  flash)  would  necessarily 
produce  the  striations.  With  a  very  fine  slit  the  coincident  D\,  D'i  or  D?,  D'% 
was  frequently  much  broadened  by  a  sort  of  burr  of  fringed  interferences. 
When  the  lines  are  self -reversed,  superposition  of  Di,  D'i,  etc.,  frequently 
showed  vivid  interferences  across  the  intensely  black  middle  line.  This  and 
the  passage  of  the  bright  and  dark  lines  across  the  superposed  D\t  D'\  lines 
of  the  solar  spectrum  are  thus  the  only  evidence  of  the  reaction  of  separated 
light-rays  on  each  other  across  an  ether  gap  observed  in  the  new  experiments, 
and  the  above  results  could  not  be  repeated. 

On  introducing  a  refined  mechanism  to  establish  the  sharpest  possible 
coincidence  of  the  DI,  D'\  or  Dz,  D'z  lines,  it  seemed  as  if  these  lines  could 
at  times  be  brought  to  overlap  with  precision,  without  the  simultaneous 
appearance  of  the  interferences  around  then;  but  on  drawing  out  the  ocular 
of  the  telescope  or  the  lens  the  cross-hatching  invariably  appears.  If  the 
coincidence  is  not  quite  sharp,  the  phenomenon  is  usually  very  strong  in  the 
isolated  bright  strip.  Horizontal  fringes  are  best  for  the  test. 

An  additional  series  of  experiments  was  made  some  time  later  by  screening 
off  parts  of  the  concave  grating  G',  in  order  to  locate  the  seat  of  the  phenom- 
enon at  the  grating.  Screening  the  transmitting  grating  G  was  without  con- 
sequence; but  on  reducing  the  area  G'  to  all  but  the  middle  vertical  strip 
about  5  mm.  wide,  a  very  marked  intensification  of  the  phenomenon  followed. 
Although  the  spectrum  as  a  whole  was  darker,  the  interferences  stood  out 
from  it,  relatively  much  sharper,  stronger,  and  broader  than  before.  The 
Fraunhofer  lines  were  still  quite  clear.  Thus  the  pattern,  g,  figure  17,  was 
now  very  common,  both  with  sunlight  and  with  sodium  light.  For  a  given 
slit  the  phenomenon  began  with  a  strong  burr  c,  figure  17,  completely  oblit- 
erating and  widening  the  superposed  D2,  D'z  lines.  When  these  lines  were 
moved  apart,  the  striations  followed  them,  as  in  figure  17,  h  and  i,  to  a  limit 
depending  on  the  width  of  the  slit.  A  still  more  interesting  pattern  is  shown 
in  figure  1 7  k,  in  which  the  interferences  proper  are  strong  and  marked  between 
the  two  Di  D'i  doublets,  but  much  fainter  striations  are  also  evident,  reaching 
obliquely  across  and  obviously  with  the  same  period. 

With  this  improvement  I  again  tested  the  ether-gap  phenomenon,  using 
the  sodium  arc,  and  to  my  surprise  again  succeeded.  D\  D\  lines  of  half 
the  breadth  of  the  doublets  apart  induced  strong  fringes  between  them,  and 
the  experiments  were  continued  with  the  same  results  for  a  long  time.  Several 
days  after,  however,  with  another  adjustment,  it  in  turn  failed.  Clearly 


32  THE   INTERFEROMETRY   OF 

there  is  some  variable  element  involved  that  escaped  me,  and  it  will  hardly 
be  worth  while  to  pursue  the  question  further  with  the  given  end  in  view, 
without  a  radical  change  of  method. 

Screening  middle  parts  of  the  grating  (in  relation  to  §  15)  did  not  lead 
to  noteworthy  results  here,  but  such  experiments  will  become  of  critical 
importance  below. 

A  word  may  be  added  in  relation  to  Fresnellian  interferences  in  the  present 
work.  These  would  be  liable  to  occur  if  the  observations  had  been  made  outside 
of  the  principal  focus,  with  the  sodium  lines  blurred.  In  all  the  experiments 
on  the  excitation  of  a  narrow  ether  gap,  however,  the  D  lines  were  clearly  in 
sight  and  sharp,  so  that  the  phenomena  of  non-reversed  spectra  and  homogene- 
ous light  (in  the  next  section)  are  not  here  in  question.  True,  such  interfer- 
ences may  often  be  found  in  the  case  of  reversed  spectra,  when  the  sodium  lines 
are  purposely  blurred,  by  pushing  the  ocular  toward  the  front  or  to  the  rear. 

12.  Experiments  continued.  Homogeneous  light. — To  turn  to  a  second 
class  of  experiments :  very  important  results  were  obtained  with  homogeneous 
light  (sodium  arc)  on  placing  the  DiD'i  or  D2Z/2  lines  in  coincidence  and  then 
broadening  the  slit  indefinitely  or  even  removing  it  altogether.  A  new  type 
of  interferences  was  discovered,  linear  and  parallel  in  character  and  inter- 
secting the  whole  yellow  field.  These  lines  could  (as  above)  be  made  to  pass 
from  a  grid  of  very  fine,  hair-like,  nearly  horizontal  lines  to  relatively  broad, 
vertical  lines,  on  changing  the  orientation  of  the  grating  Gf,  figure  16.  Small 
changes  of  position  of  the  grating  produced  a  relatively  large  rotation  and 
enlargement  of  the  lines  of  the  interference  pattern.  The  fringes,  when  verti- 
cal and  large,  are  specially  interesting.  The  distances  between  successive 
fringes  obtained  were  about  the  same  (accidentally)  as  the  DiDz  distance  of 
the  sodium  lines.  They  are  quiet  in  the  absence  of  tremor.  If  DiD'i  or  D2D'z 
were  only  present,  the  field  would  be  an  alternation  of  yellow  and  black 
striations;  but  as  both  doublets  are  present,  the  interferences  overlap  the 
flat  (non-interfering)  yellow  field  of  the  lines  not  in  coincidence.  The  fringes 
are  nevertheless  quite  distinct.  A  single  homogeneous  line  (like  the  green 
mercury  line)  would  give  better  results.  It  is  necessary  that  the  line  selected 
(say  DiD'i)  should  coincide  horizontally  and  vertically  before  the  slit  is 
broadened.  Otherwise  no  fringes  appear  in  the  yellow  ground,  or  at  least 
not  in  the  principal  focal  plane.  On  using  a  thin  mica  compensator,  it  is 
easy  to  make  these  fringes  move  while  the  mica  film  is  rotated;  and  they  pass 
from  right  to  left  and  then  back  again  from  left  to  right,  as  the  mica  vane 
passes  through  the  normal  position  of  minimum  effective  thickness.  Thus 
this  is  a  new  form  of  interferometer  with  homogeneous  light.  The  fringes 
remain  identical  in  size,  from  their  inception  till  they  vanish,  while  the  microm- 
eter M,  figure  1 6,  passes  (as  above)  over  about  15,000  wave-lengths.  In  this 
respect  the  new  interferometer  differs  from  all  other  types,  the  two  air-paths, 
GMG'  and  GNG',  alone  being  in  question.  The  condition  of  occurrence  will 
be  investigated  in  paragraph  13. 


REVERSED   AND   NON-REVERSED   SPECTRA.  33 

13.  Experiments  continued.    Contrast  of  methods. — As  these  fringes  were 
produced  with  a  concave  reflecting  grating,  the  question  may  be  put  whether 
they  would  also  appear  in  case  of  the  plane  reflecting  grating,  G',  in  the  adjust- 
ment of  figure  1 6.    The  experiment  was  therefore  repeated  with  a  wide  slit, 
or  with  no  slit  at  all,  and  there  was  no  essential  difference  in  the  two  classes 
of  results. 

On  the  contrary,  when  the  method  of  but  one  grating  and  sodium  light 
was  used  (fig.  n),  the  interferometer  fringes,  in  case  of  a  very  wide  slit  or 
the  absence  of  a  slit,  could  not  be  produced  over  the  yellow  field,  as  a  whole. 
There  appeared,  however,  an  obviously  pulsating  flicker  in  parts  of  the  field, 
on  reducing  the  width  of  the  slit  till  the  sodium  lines  were  each  about  the 
width  of  a  DiD2  space,  with  either  DJD'i  or  D^D'z  superposed.  The  sharply 
outlined  slit  showed  an  irregular,  rhythmic  brightening  and  darkening  over 
certain  parts  of  its  length.  These  broad  pulsations  were  very  violent,  very 
much  in  character  with  the  linear  phenomenon  above.  This  behavior  is  very 
peculiar;  recalling  the  appearance  of  a  bright  yellow  ribbon  undulating,  or 
flapping  fore  and  aft,  so  as  to  darken  parts  of  its  length  rhythmically.  The 
pulsations,  moreover,  were  quite  as  active  if  seen  at  night,  when  the  tremors 
of  the  laboratory  were  certainly  reduced  to  minimum.  Nevertheless,  I  am 
now  convinced  that  such  tremor  only  is  in  question. 

Regarding  the  phenomenon  as  a  whole,  one  may  argue  that  in  case  of  the 
wide  slit  and  single  grating,  in  which  the  lines  for  both  diffractions  are  there- 
fore rigorously  parallel,  the  interference  fringes  are  on  so  large  a  scale  as  to 
cover  the  whole  field  of  view  and  thus  to  escape  detection;  i.e.,  that  a  single 
vague,  quivering  shadow  of  a  flickering  field  is  all  that  may  be  looked  for, 
in  the  limited  field  of  view  of  the  eyepiece. 

Returning  to  the  case  of  two  gratings  and  the  wide  vertical  interference 
fringes  and,  in  turn,  all  but  closing  the  slit  (vertical  interferences  and  sodium 
arc  light),  the  pulsating  phenomenon  simply  narrowed  in  width.  The  two 
or  three  sharp  vibrating  lines,  alternating  in  black  and  yellow  of  the  original 
phenomenon  (Chapter  I),  did  not  appear.  The  cause  of  this  is  now  to  be 
investigated. 

14.  Experiments  continued.    Rotation,  etc.,  of  grating. — The  method  of 
two  gratings  (fig.  16  or  20,  plane  transmitting  and  concave  reflecting)  was 
first  further  improved  by  perfecting  the  fore-and-aft  motion  of  the  grating  G' 
(Gr  movable  in  the  direction  G'T  on  a  slide),  as  well  as  the  precision  of  the 
independent  rotation  of  G'  normal  to  its  face ;  i.e.,  around  G'T.    These  adjust- 
ments led  to  further  elucidation  of  the  phenomenon.    To  begin  with  the  fore- 
and-aft  motion  of  the  concave  grating  G'  (i.e.,  displacements  in  the  directions 
G'T,  fig.  20),  it  was  found  that  the  fringes,  figure  21,  a,  b,  c,  d,  e,  in  any  good 
adjustment,  pass  from  extremely  fine,  sharp,  vertical  striations,  which  gradu- 
ally thicken  and  incline  to  relatively  coarse,  horizontal  lines,  finally  with 
further  inclination  in  the  same  direction  into  fine  vertical  lines  again,  while 
G'  continually  moves  (through  about  5  cm.)  on  the  slide  normal  to  the  face 


34  THE   INTERFEROMETRY  OF 

of  the  grating.  It  was  not  at  all  difficult  to  follow  the  continuous  tilt  of  these 
lines  through  the  horizontal,  occurring  on  careful  and  continuous  front-and- 
rear  motions  of  the  grating  G'  through  the  limiting  positions.  The  fringes 
usually  vanish  vertically  merely  because  of  their  smallness. 

Again,  on  rotating  the  grating  G'  around  an  axis  normal  to  its  face,  the 
fringes  merely  vary  in  size,  without  changing  their  inclination.  Thus  if  the 
horizontal  fringes  (which  were  here  always  closer  than  the  inclined  set)  are 
in  view,  these  will  pass  from  extremely  small-sized,  fine,  hair-like  striations, 
through  a  maximum  (which  is  a  mere  shadow,  as  a  single  fringe  probably 
fills  the  field)  back  into  the  fine  lines  again.  Only  a  few  degrees  of  rotation 
of  the  grating  suffice  for  the  complete  transformation.  The  maximum  is 
frequently  discernible  only  in  consequence  of  a  flickering  field.  An  oblique 
set  of  fringes  is  equally  available,  remaining  oblique  as  they  grow  continually 
coarser  and  in  turn  finer  with  the  continuous  rotation  of  the  grating. 

When  the  very  large  horizontal  fringes  are  produced  by  this  method,  the 
change  into  vertical  fringes  by  fore-and-aft  motion  of  G'  is  very  rapid,  so 
that  relatively  wide,  nearly  vertical  forms  may  be  obtained.  All  these  effects 
may  be  produced  by  solar  or  by  arc  light,  around 

the  line  of  symmetry  of  the  overlapping  spectra;  ^/    '    '      \ 

or  with  sodium  light  when  either  DiD\  or  DJD\  21  \     //    —    ^ 

coincide.  /     """     \^ 

The  fine  vertical  or  inclined  lines  appear  as  &      fa      Qj      di 

such  when  the  slit  is  widened,  either  in  case  of 
white  or  of  sodium  light.    These  are  the  inter-  22       (     */ 
ferometer  fringes  seen  above  (§6),  coarse  or  ^     ^ 

fine.  With  sodium  light  any  width  of  slit,  or 
no  slit  at  all,  is  equally  admissible.  The  same  is  true  for  the  narrow  maxima. 
Lines  nearly  horizontal  were  sometimes  obtained,  pointing,  as  a  whole,  toward 
a  center. 

Finally  (and  this  is  the  important  result)  the  extremely  large  horizontal 
maxima,  when  a  single  fringe  fills  the  field,  can  not  be  seen  apart  from  pulsa- 
tions, in  the  case  of  a  wide  slit.  With  a  very  narrow  slit,  such  as  is  suited  for 
the  Fraunhofer  lines,  these  horizontal  fringes  appear  as  intensely  bright  or 
very  dark  images  of  the  slit.  In  other  words,  the  normal  phenomenon  of 
overlapping  symmetrical  spectra  as  described  in  Chapter  I  is  merely  the 
vertical  strip  of  an  enormous  horizontal  interference  fringe,  made  sharp  and 
differentiated  by  its  narrowness.  This  case  occurs  at  once  when  the  rulings 
of  the  two  gratings  G  and  G'  are  all  but  parallel,  and  hence  it  is  the  regular 
phenomenon  when  but  a  single  grating  is  used  for  the  two  diffractions,  as  in 
figures  ii  and  12. 

In  later  experiments  on  the  effect  of  the  rotation  of  the  grating,  G',  around 
a  normal  axis,  the  above  results  were  found  to  be  incomplete.  If  the  rotation 
is  sufficient  in  amount  (a  few  degrees,  always  very  small),  it  appears  that, 
after  enlarging,  the  fringes  also  rotate.  But  the  rotation  in  this  case  corre- 
sponds to  a  vertical  maximum,  as  indicated  in  figure  22,  the  vertical  set  being 


REVERSED   AND   NON-REVERSED   SPECTRA.  35 

the  coarsest  possible  for  a  given  fore-and-aft  position  of  the  grating  G'.  In 
the  figure,  the  sequence  a,  b,  c,  d,  e  is  obtained  for  a  continuous  rotation  of 
the  grating  (in  one  direction  around  a  normal  axis). 

It  now  became  interesting  to  ascertain  how  the  vertical  set  c,  figure  22, 
would  behave  with  the  fore-and-aft  motion.  The  experiments  showed  that 
there  was  no  further  rotation,  but  that,  while  G'  passes  normally  to  itself 
over  about  1.5  cm.  on  the  slide,  the  vertical  fringes  pass  from  extreme  fine- 
ness at  the  limit  of  visibility,  through  an  infinite  vertical  maximum  (a  single 
vague  shadow  pulsating  in  the  field) ,  back  to  extreme  fineness  again,  without 
any  rotation.  If  the  edges  of  the  corresponding  yellow  strips  (superposed 
DI,  D'\  lines)  did  not  quite  coincide,  the  fringes  were  seen  outside  of  the  prin- 
cipal focal  plane,  as  usual.  Probably  the  vertical  and  horizontal  maxima  are 
identical  in  occurrence  and  appear  in  case  of  parallelism  in  the  rulings  of  the 
two  gratings  G  and  G',  and  the  absence  of  path-difference.  Hence  if  a  single 
grating  is  used,  as  in  the  original  method,  the  interferometer  fringes  are  not 
obtainable.  This  is  an  important  and  apparently  final  result,  remembering 
that  fore-and-aft  motion  is  probably  equivalent  to  a  rotation  around  a  vertical 
axis,  parallel  to  the  grating. 

With  regard  to  the  rotation  in  case  of  fore-and-aft  motion  of  Gf,  it  is  well 
to  remark  that  in  approaching  the  position  c,  figure  24,  it  is  apt  to  be  very 
rapid  as  compared  with  the  displacement,  precisely  as  in  the  case  of  the  picket- 
fence  analogy. 

Hence  the  original  phenomenon,  consisting  of  single  lines,  can  not  be  mani- 
folded by  increasing  the  width  of  slit.  It  vanishes  for  a  wide  slit  into  an 
indiscernible  shadow.  The  phenomenon  is  a  strip  cut  across  an  enormous 
black  or  bright  horizontal  fringe,  by  the  occurrence  of  a  narrow  slit.  More- 
over, the  scintillations  variously  interpreted  above  are  now  seen  to  be  due  to 
tremors,  however  different  from  such  an  effect  they  at  first  appear;  i.e.,  the 
enormously  broad,  horizontal  fringe  changes  from  dark  to  bright,  as  a  whole, 
by  any  half  wave-length  displacement  of  any  part  of  the  apparatus.  It  is 
thus  peculiarly  sensitive  to  tremors.  On  the  other  hand,  oblique  or  fine 
vertical  fringes  are  always  recognizable  for  any  size  of  slit.  The  inquiry  is 
finally  pertinent  as  to  why  the  phenomenon  is  so  remarkably  sharpened  by 
a  narrow  slit ;  but  this  must  be  left  to  the  following  experiments. 

To  be  quite  sure  that  the  concave  grating  G'  had  no  fundamental  bearing 
on  the  phenomenon,  I  again  replaced  it  by  the  Michelson  plane  reflecting 
grating  (fig.  16,  G  transmitting,  G'  reflecting).  In  the  same  way  I  was  able 
to  rotate  the  fringes,  continuously,  through  a  horizontal  maximum  of  size 
by  fore-and-aft  motion  of  G'.  Rotation  of  G'  in  its  own  plane  increased  or 
decreased  the  breadth  and  distance  apart  of  the  fringes  through  a  maximum, 
coinciding  with  the  parallelism  of  the  rulings  of  the  two  gratings.  Here  I 
also  showed  decisively  that  as  the  rungs  of  the  interference  ladder  (fig.  2 1  c) 
thickened  and  receded  from  each  other,  the  design  passed,  in  the  transitional 
case,  through  the  original  phenomenon  of  the  single  vertical  line  dark  or 
brilliant  yellow,  for  a  slit  showing  the  Fraunhofer  lines  clearly.  The  phenome- 


36  THE   INTERFEROMETRY   OF 

non  vanishes  with  the  spectrum  lines  as  the  slit  is  widened,  but,  on  the  other 
hand,  persists  as  far  as  the  interference  of  light  for  a  narrow  slit.  Finally, 
the  apparent  occurrence  of  more  than  one  line  is  referable  to  the  presence  of 
more  than  one  nearly  horizontal  wide  band  in  the  field  of  the  telescope.  Thus, 
for  instance,  cases  between  b  and  c  near  c  and  between  c  and  d  near  c,  figure  24, 
are  the  ones  most  liable  to  occur  when  both  diffractions  take  place  at  a  single 
grating.  This  result  will  be  used  in  paragraph  15. 

15.  Tentative  equations. — In  the  first  place,  the  actual  paths  (apart  from 
the  theory  of  diffraction)  of  the  two  component  rays,  on  the  right  and  left 
sides  of  the  line  of  symmetry,  II' Z,  figure  23,  will  be  of  interest.  The  compu- 
tation may  be  made  for  the  method  of  two  gratings  at  once,  as  the  result 
(if  the  distance  apart  of  the  gratings  is  C  =  o)  includes  the  method  with  one 
grating;  i.e.,  the  more  complicated  figure  23,  where  G  is  the  transmitting  and 
G'  the  reflecting  grating,  resolves  itself  into  a  case  of  figure  24,  with  but  one 
grating,  G.  M  and  M'  are  the  two  opaque  mirrors,  I  the  normally  incident 
homogeneous  ray.  Supposing,  for  simplicity,  that  the  grating  planes  G  and 
G'  are  parallel  and  symmetrically  placed  relatively  to  the  mirrors  M  and  M' , 
as  in  the  figure,  the  ray  Y  diffracted  at  the  angle  0i  is  reflected  into  X  at  an 
angle  82-61  and  diffracted  into  Z  normally,  at  an  angle  0%,  on  both  sides. 
Under  the  condition  of  symmetry  assumed  X+Y—  (X'+Y'}=o,  or  without 
path-difference.  Let  N  be  the  normal  from  I  to  M,  and  n  the  normal  from 
I'  to  M,  with  a  similar  notation  on  the  other  side.  Hence  if  I  be  given  an 
inclination,  di,  61  is  incremented  by  d0\,  Y-\-X  passes  into  y\-\-y-\-x,  Y'-\-X' 
into  y'\+y'+xr,  decremented  at  an  angle  dB'i,  while  both  are  diffracted  into 
Z'.  Since  generally 


sm  0i-sm*  =  X/.L>  cos  0id0i  =  cos  0  i  d8  i 

for  homogeneous  light  and  the  same  di.    Hence  ddi  =  dd'i=dd,  say. 
If  5=  02—  0i,  and  <t=  02+0i,  the  auxiliary  equations 

~sin  0i-f-sin  02  r_    N—n 

sin  5  cos  (<r/2) 

are  useful.     From  a  consideration  of  the  yC  and  y*  triangles,  moreover, 
the  relations  follow: 


v_ 

cos  (8/2 -de)  yi     cos(<7/2)cos(01-r-d0)  '    ycos(92-d6) 

and  from  the  y\C  and  y'x  triangles,  similarly, 

/  ,    ,  =          N  ,  = N—n ,      ,cos(0i-c/0) 

1    cos  (3/2-M0)  cos  (er/2)  cos  (01-d0)  ~y  cos  (02+d0) 

Hence,  after  some  reduction,  the  path  on  one  side  is 

2)  N-n 


cos  (02-d0)     cos  (<r/2)  cos  (Ot-dff) 


REVERSED   AND   NON-REVERSED    SPECTRA. 


37 


which  may  be  further  simplified  to 


N  coscr+n 


COS  (62  —  d6)  COS((T/2) 

From  this  the  path  on  the  other  side  will  be 

,,    ,.    ,  _  N  cos  <r+n 


The  path-difference,  AP,  thus  becomes,  nearly, 


Ap_>rcosa-+n/ 


cos  (<r/2)\cos  (02+<f0)     cos(02-d0) 


i          \ 
2-d0)7 


JVcosq-+n  2  sin  02 
cos  (<r/2)    cos2  8* 


This  is  perhaps  the  simplest  form  attainable.  If,  apart  from  diffraction, 
this  should  result  in  interference,  the  angular  breadth  of  an  interference 
fringe  would  be  (AP  =  X) 


X  cos2  62    cos 


2  sin  02  N  cos  ff-\-n 
and  if  D  is  the  grating  space  and  sin  0  =  X/D', 

de  =  (£>'2~X2)  cos  (a/2} 
2D'(Ncos<r+n) 

25     ^ 


^' 


or 


In  case  of  a  single  grating 

0-/2  =  02=0  Af  =  W 

aAT  cos2  0  2  sin  0 


cos  a  =  2  cos2  0—i 


cos  0       cos2  0 

a  result  which  may  be  reduced  more  easily  from  figure  24.    Hence,  the 
angular  distance  apart  of  the  fringes  would  be  (AP  =  X) 

X  D  cos  0    \/£>2-X2 

dd= = = i^ — 

4N  tan  0        4N  *N 

if  D  is  the  grating  space.    To  find  the  part  of  the  spectrum  (dX)  occupied 
by  a  fringe  in  the  case  postulated,  since  sin  0  =  \/D, 

dd  d\      ^D 

cos  0    Z)  cos2  0    4A/' 


86413 


38  THE    INTERFEROMETRY   OF 

and  from  the  preceding  equations,  finally, 


where  dK  would  be  the  wave-length  breadth  of  the  fringe,  remembering  that 
the  fringes  themselves  are  homogeneous  light. 
In  the  grating  used 

"8  cm.  X  =  6oXio~6cm.  w=ioocm. 


400 

This  is  but  1/200  of  the  distance,  dX  =  6Xicr8  cm.,  between  the  D  lines. 
Hence  such  fringes  would  be  invisible.  Moreover,  d6  oc  i/N;  the  fringes,  there- 
fore, should  grow  markedly  in  size  as  AT  is  made  smaller.  Experiments  were 
carried  out  with  this  consideration  in  view,  by  the  single-grating  and  concave- 
mirror  method,  N  being  reduced  from  nearly  2  meters  to  20  cm.,  without  any 
observable  change  in  the  breadth  or  character  of  the  phenomenon.  It  showed 
the  same  alternation  of  one  black  and  one  or  two  bright  linear  fringes,  or 
the  reverse,  throughout.  Hence,  it  seems  improbable  that  the  phenomenon, 
i.e.,  the  interference  fringes,  are  referable  to  such  a  plan  of  interference  as 
is  given  in  figure  24. 
Similarly,  for  the  case  of  two  gratings,  figure  23, 

COS  Qr/2)  (D'2-X2) 
2D'(N  COS  <r+ri) 

where,  if  we  insert  the  data  0i  =  9°4o'  and  02=  i9°5S' 

D'=i73Xio~6  cm.  N=i62  cm.  w  =  82cm.  X  =  s8.9Xio~6  cm. 

then 

,.  .967Xio-10X263  ,      ,. 

di=  .^X346X(.6aX.87+8.)  =  -33  X  ^  radmns. 

Thus  dd  is  about  of  the  same  small  order  of  values  as  above,  i.e.,  less  than 
one-tenth  second  of  arc  or  i/iooo  of  the  DiD2  space,  and  thus  quite  inappre- 
ciable. Some  other  source,  or  at  least  some  compensation,  must  therefore 
be  found  for  the  interferometer  interferences  seen  with  homogeneous  light. 

The  full  discussion  of  the  effective  path-difference  in  terms  of  the  diffrac- 
tions occurring  will  be  given  in  §  27,  in  order  not  to  interrupt  the  progress 
of  the  experimental  work  here.  It  will  then  be  obvious  that  the  mere  effect 
of  changing  the  obliquity  of  the  incident  homogeneous  rays,  I,  introduces 
no  path-difference,  or  that  the  fringes  observed  are  varied  by  the  displace- 
ments and  rotations  of  the  grating,  Gr,  and  the  mirrors  M  and  N. 

16.  Experiments  continued.  Analogies.—  With  this  possible  case  disposed 
of,  it  now  becomes  necessary  to  inquire  into  the  other  causes  of  the  phenome- 
non, as  described  in  paragraph  13.  This  is  conveniently  done  with  reference 
to  figure  26,  where  n  and  n'  are  the  axes  of  the  pencil  of  yellow  light,  reflected 
from  the  opaque  mirrors  M  and  N,  after  arrival  from  the  transmitting  grating 


REVERSED   AND   NON-REVERSED   SPECTRA. 


39 


G.  It  is  necessary  to  consider  the  three  positions  of  the  reflecting  grating  G'; 
viz,  G',  G'i,  and  G'z.  In  the  symmetrical  position  G',  the  pencils  whose  axes 
are  n  and  n'  meet  at  a  and  are  both  diffracted  along  r.  In  the  position  G', 
they  are  separately  diffracted  at  b  and  b'  in  the  direction  r\  and  r'i,  and  they 
would  not  interfere  but  for  the  objective  of  the  telescope,  or,  in  the  other 
case,  of  the  concave  mirror  of  the  grating.  In  the  position  G'z,  finally,  the 
pencils  n  and  n'  are  separately  diffracted  at  c  and  c'  into  r2  and  r'z  and  again 
brought  to  interference  by  the  lens  or  concave  mirror,  as  specified. 

Now  it  is  true  that  the  rays  na  and  n'a  (position  G'),  though  parallel  in  a 
horizontal  plane,  are  not  quite  collimated  in  a  vertical  plane.  The  pencils 
are  symmetrically  oblique  to  a  central  horizontal  ray  in  the  vertical  plane, 
and  their  optical  paths  should  therefore  differ.  But  fringes,  if  producible  in 
this  way  here,  have  nothing  to  do  with  the  rotation  of  the  grating  in  its  own 
plane  and  may  here  be  disregarded,  to  be  considered  later. 


26       c/fc  27 

To  take  the  rotation  of  the  fringes  first,  it  is  interesting  to  note  in  passing 
that  the  interferences  obtained  by  rotation  around  a  normal  axis  recall  the 
common  phenomenon  observed  when  two  picket  fences  cross  each  other  at  a 
small  angle  <p.  It  may  therefore  be  worth  while  to  briefly  examine  the  rela- 
tions here  involved  (fig.  27)  where  5'  and  5  are  two  corresponding  pickets  of 
the  grating  at  an  angle  <p  and  the  normals  D'  and  D  are  the  respective  grating 
spaces.  The  intersections  of  the  groups  of  lines  5'  and  5  make  the  representa- 
tive parallelogram  of  the  figure  (5  taken  vertical),  of  which  B  is  the  large 
and  B'  the  small  diagonal.  The  angles  indicated  in  the  figure  are  x+y=<p 
and  x'+y'+<p=  180°.  As  the  bright  band  in  these  interferences  is  the  locus 
of  the  corners  in  the  successive  parallelograms,  B  is  the  distance  between 
two  bright  bands,  while  B',  making  an  angle  y'  with  5,  is  the  direction  of 
these  parallel  interference  bands  relative  to  the  vertical.  Let  the  free  ends 
of  D  and  D'  be  joined  by  the  line  Ef;  and  if  D  is  prolonged  to  the  left  and  the 
intercept  is  D  in  length,  let  this  be  joined  with  the  end  of  D'  by  E.  Then  the 
triangle  DED'  and  S'BS,  DE'D'  and  S'B'S,  may  be  shown  to  be  similar  by 
aid  of  the  following  equations : 

SD  =  S'D'  D' sm<p  =  Esinx  S'  sin  <f>=B  siny 


D' 
S7 


E  sinx 
B  siny 


40  THE   INTERFEROMETRY   OF 

If  E  is  expressed  in  terms  of  D  and  D',  and  B  in  terms  of  5  and  S',  and 
the  first  equation  is  used,  then 

5 


S'     sin  y     D 

from  which,  in  the  fourth  equation, 
E 


sin  <p                    sin  <p 
Similarly,  


B,  =  E'  =  VD*+D'2-  2DD'  cos  y 
sin  ^>  sin  v> 

Again,  the  angle  y'  is  given  from 


sin  y'  D 

or  on  reduction 

,  _     D  sin  <p 

te*y-D,_DcoS(f> 

KD=D',  or  5=5',  then 

D 


sin  (?/a)  cos  (>/a) 

tan  y'  =  .  v  =  cos  (<p/2)/sin  (^/a),  or  tan  y'  tan  ^/a  =  i 

sin  (v/2) 

Thus  if  <p  =  o,  tany  =  oo,  y'  =  go  ,  or  the  fringes  are  horizontal  and  B  =  zS. 
If  y'  is  nearly  zero 


changing  very  rapidly  with  <p. 

If  one  grating  of  a  pair,  with  identical  grating  spaces  D,  is  moved  parallel 
to  itself,  in  front  of  the  other,  the  effect  to  an  eye  at  a  finite  distance  is  to 
make  the  grating  spaces  D  virtually  unequal;  or 


j \ 

2  COS  (<f>/2) 

so  that  for  an  acute  angle  <p,  the  fringe  breadth  is  increased.  Thus  BQ  is  a 
minimum  in  case  of  coincident  gratings. 

The  analogy  is  thus  curiously  as  follows:  The  fringes  just  treated  rotate 
with  the  rotation  of  either  grating  in  its  own  plane  and  pass  through  a  mini- 
mum size  with  fore-and-aft  motion;  whereas  in  the  above  results  the  optical 
grating  showed  a  passage  through  a  maximum  of  size  with  the  rotation  of 
either  grating  in  its  own  plane  and  a  rotation  of  fringes  with  fore-and-aft 
motion  of  the  grating. 

Returning  from  this  digression  to  figure  26,  if  the  grating  G'  is  not  quite 
symmetrical,  but  makes  a  small  angle  <p  with  the  symmetrical  position  as  at 
g',  the  fore-and-aft  motion  will  change  the  condition  of  path-excess  on  the 
right  (position  g'2,  M  path  larger)  to  the  condition  of  path-excess  on  the  left 


REVERSED   AND   NON-REVERSED   SPECTRA.  41 

(position  g'i,  N  path  larger) ;  and  if  the  motion  is  continuous  in  one  direction, 
g'-i,  g',  g'a,  the  path-difference  will  pass  through  zero.  No  doubt  the  angle 
<p  is  rarely  quite  zero,  so  that  this  variable  should  be  entered  as  an  essential 
part  of  the  problem.  The  resulting  conditions  are  complicated,  as  there  are 
now  two  angles  of  incidence  and  diffraction  and  it  will  therefore  be  considered 
later  (§  28).  It  is  obvious,  however,  that  if  for  a  stationary  grating  £', 
figure  26,  the  angle  <p  is  changed  from  negative  to  positive  values,  through 
zero,  the  effect  must  be  about  the  same  as  results  from  fore-and-aft  motion. 
In  both  cases  excess  of  optical  path  is  converted  into  deficiency,  and  vice  versa. 
Hence,  as  has  been  already  stated,  the  effects  both  of  the  fore-and-aft  motion 
and  of  the  rotation  of  the  grating  G'  around  a  vertical  axis  parallel  to  its 
face  conform  to  the  interference  fringes  of  figure  21,  a  to  e. 

It  is  common,  moreover,  if  a  concave  grating  is  used  (with  parallel  rays) 
at  G',  to  find  the  two  sodium  doublets  due  to  reflection  from  M  and  N  ap- 
proaching and  receding  from  each  other  in  the  field  of  view  of  the  ocular 
when  the  grating  G'  is  subjected  to  fore-and-aft  motion.  This  means  that 
although  the  axes  of  incident  rays  are  parallel  in  two  positions,  whenever  i 
varies  (as  it  must  for  a  concave  grating  and  fore-and-aft  motion),  the  diffracted 
rays  from  M  and  N  do  not  converge  in  the  same  focus  in  which  they  originally 
converged,  but  converge  in  distinct  foci.  For  if  sin  * — sin  6  =  \/D  or  cos  idi  = 
cos  0d6,  suppose  that  for  a  given  i,  0  =  o;  then  cosidi=dd.  But  the  devia- 
tion, 5,  of  the  diffracted  ray  from  its  original  direction  is  now  di+d0,  or 

8=di(i  +cos  i)  =  zdi  cos2  i/2 

Similarly,  the  principal  focal  distance  p',  for  varying  i,  is  not  quite  constant. 
From  Rowland's  equation,  if  parallel  rays  impinge  at  an  angle  i  and  are 
diffracted  at  an  angle  0=o, 

/=      R      =       R 
p      i  +  cost     2  cos2 i/2 

If  i=zo°,  then  cos2  i/2  =  .976,  and  p'=R/2,  nearly  but  not  quite. 

I  have  not  examined  into  the  case  further,  as  both  the  sodium  doublets 
are  distinctly  seen  if  the  ocular  follows  them  (fore  and  aft),  and  the  lateral 
displacement  of  doublets  is  of  minor  interest. 

With  the  plane  reflecting  grating  this  discrepancy  can  not  enter,  since  for 
parallel  rays  the  angles  of  incidence  remain  the  same  throughout  the  fore-and- 
aft  motion,  and  therefore  the  arigles  of  diffraction  would  also  be  identical. 

Two  outstanding  difficulties  of  adjustment  have  still  to  be  mentioned, 
though  their  effect  will  be  discussed  more  fully  in  the  next  chapter.  These 
refer  to  the  rotation  of  the  grating  G',  around  a  vertical  axis  and  around  a 
horizontal  axis,  in  its  own  plane  or  parallel  to  it.  The  rotation  around  the 
vertical  axis  was  taken  up  in  a  restricted  way  above,  in  figure  13,  Chapter  II. 
The  effect  (rotation  of  G')  is  to  change  the  inclination  of  the  fringes  passing 
from  inclination  to  the  left  through  zero  to  inclination  towards  the  right. 
The  effect  is  thus  similar  to  the  fore-and-aft  motion,  as  shown  in  figure  21. 


42  THE   INTERFEROMETRY   OF 

It  was  here,  with  the  ocular  thrown  much  to  the  right  (near  M),  that  I 
again  encountered  the  arrow-shaped  fringes  of  figure  2,  D,  Chapter  I.  Though 
they  are  rarely  quiet,  the  observation  can  not  be  an  illusion.  As  seen  with 
white  light  and  a  fine  slit  they  are  merely  an  indication  of  fringes  which,  when 
viewed  with  a  broad  slit  and  homogeneous  light,  will  be  horizontal. 

Rotation  around  a  horizontal  axis  parallel  to  the  face  of  the  grating  must 
also  destroy  the  parallelism  of  the  rulings.  The  usual  effect  was  to  change  the 
size  of  fringes  (distance  apart,  etc.)  ;  but  I  was  not  able  to  get  any  consistent 
results  on  rotating  G  ',  owing  to  subsidiary  difficulties.  On  rotating  the  grating 
G,  however,  a  case  in  which  fine  rotation  around  a  horizontal  axis  was  more 
fully  guaranteed,  the  fringes  passed  with  continuous  rotation  through  a 
vertical  maximum,  as  in  figure  22. 

In  figure  26,  the  central  region  a  of  the  grating  G'  is  found,  on  inspection, 
to  be  yellow  in  the  position  G',  red  in  the  position  G'\,  and  green  in  the  posi- 
tion G'z-  The  slit  in  this  case  must  be  very  fine.  For  a  wide  slit  and  homo- 
geneous light,  the  continuous  change  in  the  obliquity  of  pencils  is  equivalent 
to  the  continuous  change  of  wave-length  in  the  former  case.  It  is  therefore 
interesting  to  make  an  estimate  of  the  results  to  be  expected,  if  the  vertical 
fringes  for  the  cases  bb'  or  ccf  were  Fresnellian  interferences,  superposed  on 
whatever  phase-difference  arrives  at  these  points.  In  the  usual  notation,  if 
c  is  the  effective  width  at  the  concave  grating,  F  its  principal  focal  distance, 
x  the  deviation  per  fringe,  dd  the  corresponding  angle  of  deviation,  X  the 
wave-length  of  light, 

x  =  \F/cord8  =  \/c 

If  c=i.6  cm.,  X  =  6Xicr6  cm.,  then  d0  =  3.7Xicr5,  or  about  T"  of  arc. 

The  corresponding  deviation  ddD  equivalent  to  d\D  of  the  DiDz  lines  would 
be  (if  the  grating  space  is  D=  17  3X10"*  cm.,  0  =  20°,  nearly,  the  normal 
deviations  for  yellow  light), 

,,          d\D  6XIO-8 


Thus  ^0-  =  o.i,  or,  in  this  special  case,  there  would  be  ten  hair  lines  to  the 

DiD2  space.  As  c  is  smaller  or  larger,  there  would  be  more  or  less  lines. 
This  is  about  the  actual  state  of  the  case  as  observed.  Finally,  if  c  is  very 
small,  the  fringes  are  large,  since 

de  =X£>cos  6 
deD~    cd\D 

Thus  conditions  for  practical  interferometry  would  actually  appear,  the 
fringes  being  of  DiD2  width  for  c  =  0.17  cm.,  provided  a  wide  slit  and  homo- 
geneous light  of  at  least  Di  or  Dz  grade  is  used.  Such  an  interferometer 
seems  to  differ  from  other  forms,  inasmuch  as  the  fringes  remain  of  the  same 
size  and  distribution,  from  their  entrance  into  the  field  to  their  exit;  or  for  a 
motion  of  the  opaque  mirror  M  of  about  6  mm. 


REVERSED   AND   NON-REVERSED   SPECTRA.  43 

To  resume  the  evidence  thus  far  obtained,  we  may  therefore  assert  that 
in  the  case  of  homogeneous  light  and  a  wide  slit,  or  the  absence  of  a  slit,  the 
field  would  either  be  bright  or  dark,  as  a  whole.  There  is  a  single  enormous 
horizontal  fringe  in  the  field.  Hence  the  pronounced  flickering  with  half 
wave-length  displacements  of  any  part  of  the  apparatus.  With  the  slit 
narrowed  until  the  Fraunhofer  lines  are  seen  sharply,  the  linear  phenomenon 
in  question  (Chapter  I)  appears.  This  may  become  ladder-like,  but  it  always 
remains  very  narrow  (^  DiDz)  when  the  rulings  of  the  two  gratings  are  not 
quite  parallel. 

17.  Subsidiary  diffractions. — The  behavior  of  the  linear  phenomena  some- 
times suggests  probable  relations  to  the  Fresnellian  interferences,  produced, 
however,  not  within  the  telescope,  as  in  §§  27,  28,  Chapter  III  (for  the  inter- 
ferences are  seen  together  with  the  Fraunhofer  lines  in  the  principal  focal 
plane),  but  outside  of  it,  at  the  grating,  as  suggested  by  figure  26.  If  the 
concave  grating  G'  is  screened  off,  until  a  width  of  strip  parallel  to  the  rulings 
and  not  more  than  5  mm.  wide  is  used,  the  linear  phenomenon  is  much  en- 
hanced, being  both  broader  and  stronger,  without  losing  its  general  character. 
Here  the  D  lines  are  still  visible.  The  ladder-like  patterns  show  an  equally 
pronounced  coarsening.  So  far  as  these  phenomena  go,  it  is  obvious  that  the 
resolving  power  of  the  grating  must  be  in  question,  seeing  that  the  total 
number  of  rulings  has  been  greatly  reduced.  The  use  of  screens  with  narrower 
slits  carries  the  process  farther;  but  after  the  opening  is  less  than  2  mm.  in 
width  the  available  light  is  insufficient  for  further  observation.  If  a  small 
lens  is  used,  the  phenomena  can  still  be  seen  over  2  meters  beyond  the  principal 
focus  of  the  grating. 

A  screen  was  now  made  as  in  figure  28  a,  with  two  slits  about  2  mm.  wide 
and  2  mm.  apart  (6),  and  placed  over  the  effective  part  of  the  grating.  The 
result,  after  careful  trial  as  to  position,  was  noteworthy.  Oblique  fringes 
were  widened  to  many  times  the  DiD2  space  and  coarsened,  showing  a  definite 
grid-like  design,  as  in  figure  28  b,  whereas,  on  removing  the  screen,  the  original 
pattern  of  a  regular  succession  of  brilliant  dots  (fig.  28  c)  again  appeared. 

It  was  with  the  linear  fringes,  however,  that  the  evidence  obtained  was 
most  striking;  for  these  now  showed  all  the  Fresnellian  interferences  (fig.  28  d). 
On  removing  the  screen,  the  brilliant  linear  phenomenon  (fig.  28  e),  which 
in  all  the  experiments  made  had  thus  far  resisted  manifolding,  appeared  at 
once.  The  pattern,  d,  moreover,  when  viewed  with  a  small  lens,  within  a 
meter  in  the  direction  of  the  rays,  showed  very  definite  enlargement  with 
distance.  Though  a  fine  slit  was  needed,  the  resolving  power  of  the  grating 
was  now  too  small  to  show  any  Fraunhofer  lines.  Similar  results  were  obtained 
for  a  wire  i  or  2  mm.  in  diameter.  With  the  screen,  figure  28  a,  and  a  bar,  6, 
i  mm.  wide,  the  fine  interference  grid  due  to  the  bar,  and  the  coarse  grid  due 
to  the  spaces  (the  fine  lines  being  about  twice  as  narrow  as  the  coarse,  but 
all  of  the  same  inclination)  were  often  obtained  together  (fig.  28/).  A  space 
i  cm.  wide  intersected  by  a  bar  2  mm.  wide  gave  similar  results,  fine  grids  or 


44 


THE   INTERFEROMETRY   OF 


thick  lines,  according  as  one  or  both  spaces  were  used.  If  either  mirror,  M 
or  N,  is  screened,  the  whole  phenomenon  vanishes. 

It  follows,  then,  if  rv  and  v'r',  figure  29,  represent  the  two  reversed,  over- 
lapping spectra  at  the  grating,  b  the  focus  and  aa'b  the  direction  of  the  homo- 
geneous diffracted  rays  condensed  at  6,  that  about  0.5  cm.  of  the  spectrum,  d'a' 
and  ad  on  either  side  of  a,  is  chiefly  active  in  modifying  the  resulting  diffrac- 
tion pattern.  Within  this  the  homogeneous  rays,  cc'  and  dd',  are  capable  of 
interference.  Although  the  wave-fronts  entering  b  are  slightly  spherical,  their 
radius  is  about  r=i  meter,  and  they  may  therefore  be  regarded  plane.  In 
such  a  case  the  angular  width  dx  of  the  illuminated  strip  at  b,  for  a  width  of 
screen  dd' =  i  cm.,  between  two  extinctions,  may  be  written 
to  X |  =  6oX^  =  6><io.5 
r  dd  i 

whereas  the  angular  breadth  of  the  DiD2  doublets  is  about  37Xicr5;  i  e., 
the  rays  from  d  and  d' ,  if  in  phase,  should  cease  to  illuminate  &  at  a  breadth 
of  about  one-sixth  the  distance  between  the  sodium  lines.  The  rays  within 


o  d   e,f 


29 


dd'  would  correspond  to  greater  widths;  those  from  cc',  for  instance,  0.5  milli- 
meter apart,  would  illuminate  twice  the  estimated  width,  so  that  a  strip  at 
6,  with  a  breadth  of  one-third  the  interval  DiDz,  is  a  reasonable  average.  All 
rays,  however,  would  produce  illumination  at  b.  As  the  screens  are  nar- 
rower, not  only  would  the  fringe  be  broader,  but  more  lines  would  appear, 
because  there  is  less  overlapping.  All  this  is  in  accord  with  observation. 
Excepting  the  occurrence  of  independent  half  wave-fronts,  the  phenomena 
do  not  differ  from  the  ordinary  diffraction. 

With  regard  to  waves  of  slightly  different  lengths,  focussed  at  &',  each  is 
there  superposed  on  a  wave  of  different  length  from  its  own,  and  appreciable 
interference  ceases  for  this  reason.  If  the  slit  is  widened,  the  phenomenon 
(with  white  light)  also  vanishes  by  overlapping.  The  case  of  the  screen  with 
two  spaces  has  already  been  treated  in  relation  to  figure  26.  In  general,  these 
are  cases  of  the  diffraction  of  a  rod,  or  of  a  slit,  which  are  possible  only  if  the 
colors,  X,  are  symmetrically  distributed  to  the  right  and  to  the  left  of  it.  Thus 
they  require  both  spectra  and  can  not  appear  if  single  spectrum  only  is  present. 
To  reveal  the  nature  of  the  phenomenon,  a  wide  slit  and  homogeneous  light 
must  be  resorted  to,  as  has  been  done  in  the  present  paper,  even  if  white  light 
and  the  fine  slit  totally  change  the  aspect  of  the  fringes. 


REVERSED   AND   NON-REVERSED   SPECTRA.  45 

18.  Conclusion. — To  return,  finally,  to  the  original  inference,  it  appears 
that  beating  wave-trains  have  not  been  observed,  but  that  the  striking  scin- 
tillations are  due  to  an  exceptional  susceptibility  of  the  apparatus  to  labora- 
tory tremors,  when  exhibiting  the  phenomenon  in  question.  Of  this  I  further 
assured  myself  by  observations  made  at  night  and  on  Sunday,  though  there  is 
some  doubt  in  my  mind.  What  has  certainly  been  observed  is  the  inter- 
ference of  a  D\  or  Dz  line  with  a  reversed  D'\  or  J9'2  line,  both  having  the  same 
source  and  longitudinal  axis.  One  can  only  assert,  therefore,  that  light  of 
the  wave-length  interval  of  the  breadth  of  these  lines  is  capable  of  interference, 
when  the  line  is  reversed. 

The  phenomena,  as  a  whole,  are  to  be  treated  as  diffractions  of  symmetrical 
half  wave-fronts,  each  of  which  may  be  separately  controlled  by  the  corre- 
sponding micrometer. 


CHAPTER  III. 


THE  INTERFERENCES  OF  THE  NON-REVERSED  SPECTRA  OF  TWO  GRATINGS, 
TOGETHER  WITH  AN  INTERPRETATION  OF  THE  PHENOMENA  IN 
CHAPTERS  I  AND  H. 

19.  Introduction.  Method. — The  chief  purpose  of  the  present  paper  is 
the  search  for  phenomena  similar  to  those  of  Chapter  II,  but  in  which  the 
two  spectra  brought  to  interference  are  not  inverted  relatively  to  each  other. 
Incidentally  the  strong  interferences  may  have  a  value  on  their  own  account. 
It  has  been  shown  that  the  totality  of  the  phenomena  with  spectra  reversed 
on  a  transverse  or  a  longitudinal  axis  are  quite  complicated,  and  a  series  of 
companion  researches  in  which  similar  results  are  aimed  at,  in  the  absence 
of  inversion,  is  thus  very  desirable. 

The  apparatus  (fig.  30)  is  a  modification  of  that  shown  in  figure  50,  in  the 
next  section,  MM  being  the  base  of  the  Fraunhofer  micrometer,  55  the  slide, 
E  the  micrometer  screw.  The  brass  capsules  A  and  B  are  securely  mounted 
on  the  slide  5,  free  from  the  base  M,  and  on  the  base  M  free  from  the  slide 


5,  respectively.  Each  capsule  is  provided  with  three  adjustment  screws 
relative  to  horizontal  and  vertical  axes  a,  b,  br,  and  c,d,d',  together  with  strong 
rearward-acting  springs,  by  which  the  gratings  G  and  H  at  a  distance  e  apart 
may  each  be  rotated  slightly  around  a  vertical  or  a  horizontal  axis  (plane  dot 
slot  mechanism) .  The  two  gratings  G  and  H  must  be  identical,  or  very  nearly 
so,  as  to  the  number  of  lines  per  inch,  and  with  their  ruled  faces  toward  each 
other.  These  faces,  as  well  as  the  ruled  lines,  are  to  be  nearly  in  parallel. 
To  secure  the  latter  adjustment  a  bolt,  g,  normal  to  the  face  of  the  grating 
H,  serves  as  an  axis,  and  an  available  tangent  screw  and  spring  (not  shown) 
is  at  hand  for  fine  adjustment.  This  device  is  of  great  importance  in  bringing 
the  longitudinal  axes  of  the  two  spectra  due  to  G  and  H  into  coincidence, 
and  a  fine  wire  must  be  drawn  across  the  slit  of  the  collimator  to  serve  as  a 
guiding-line  through  the  spectrum.  Any  lack  of  parallelism  in  slit  and  rulings 
rotates  the  fringes. 
46 


REVERSED   AND   NON-REVERSED   SPECTRA.  47 

The  beam  of  light,  L,  either  white  or  homogeneous,  as  the  experiment  may 
require,  is  furnished  by  a  collimator  (not  shown),  which,  with  the  telescope  at 
T  (placed  in  plan,  in  figure  31,  at  T  or  D),  are  the  usual  parts  of  a  spectro- 
scope. The  collimator  with  slit  is  always  necessary  for  adjustment.  It  may 
then  be  removed  if  the  phenomenon  is  to  be  studied  in  the  absence  of  the 
slit.  The  telescope  is  frequently  replaced  to  advantage  by  a  lens.  White 
light  is  to  be  furnished  by  the  arc  lamp  (without  a  condenser),  by  sunlight, 
or  by  an  ordinary  Welsbach  burner.  Both  spectra  are  naturally  very  intense. 
A  sodium  flame  suffices  for  the  work  with  homogeneous  rays. 

The  adjustments  in  case  of  white  light  are  simple  and  the  interferences 
usually  very  pronounced,  large,  and  striking.  Brilliant  spectra,  channeled 
with  vertical  narrow  black  lines,  are  easily  obtained  when  the  longitudinal 
axes  are  placed  accurately  in  coincidence  by  rotating  the  plate  h  carrying 
the  grating  H,  on  the  plate  /,  around  the  axis  g.  If  the  gratings  are  quite 
identical  the  sodium  lines  will  also  be  in  coincidence.  Otherwise  the  two 
doublets,  DiDz  and  D'\D'z,  of  the  two  spectra  (nearly  identical  in  all  their 
parts  and  in  the  same  direction)  are  placed  in  coincidence  by  rotating  either 
grating  around  a  vertical  axis.  Thereupon  the  strong  fringes  will  usually  ap- 
pear for  all  distances,  e,  less  than  2  cm.  These  fringes  are  nearly  equidistant 
and  vertical  and  intersect  the  whole  spectrum  transversely.  They  are  not 
complicated  with  other  fringes,  as  in  the  experiments  of  the  next  section. 
They  increase  in  size  till  a  single  shadow  fills  the  field  of  view,  in  proportion 
as  the  distance  e  is  made  smaller  and  smaller  to  the  limit  of  complete  contact. 
With  the  two  adjustments  carefully  made,  finally,  by  aid  of  the  fringes 
themselves,  further  trials  for  parallelism  are  not  necessary.  Two  film  grat- 
ings, or  even  films,  give  very  good  fringes.  During  manipulations  great  care 
must  be  taken  to  keep  the  angle  of  incidence,  i,  rigorously  constant;  i.e., 
to  avoid  rotating  both  gratings  together  or  the  apparatus  as  a  whole,  as  this 
displaces  the  sodium  doublets  relative  to  each  other  and  seriously  modifies 
the  equations. 

20.  White  light.  Colored  fringes. — The  two  sodium  doublets  seen  in  the 
arc  spectrum  are  usually  equally  brilliant,  and  but  one  set  of  strong  fringes 
is  present  in  the  field  of  the  telescope.  Relatively  faint  fringes  may  some- 
times occur,  due,  no  doubt,  to  reflection,  as  investigated  in  the  next  section. 

If  both  gratings  are  rotated,  changing  the  angle  of  incidence  from  o°  to  «°, 
the  fringes  disappear  from  the  principal  focal  plane,  but  reappear  strongly 
in  another  focal  plane  (ocular  forward  or  rearward).  In  such  a  case  the  D 
lines  are  no  longer  superposed.  To  be  specific,  let  i  and  i',  6  and  6',  be  the 
angles  of  incidence  and  diffraction  at  the  two  gratings  in  question,  the  angle 
between  their  ruled  faces  being  i-i' .  Let  D  and  D'  be  the  two  grating  con- 
stants, and  nearly  equal.  Then  for  a  given  color,  X,  in  relation  to  the  individual 
normals  of  the  two  gratings, 

sin  6  -  sin  i  =  \/D  sin  tf  -  sin  i' = \/D' 


48  THE   INTERFEROMETRY   OF 

Now  if  6'  is  referred  to  the  original  normal  it  becomes  0"=  0'+t—  *', 

or  sin  (  0"  -  *  +»')  -  sin  i  =  \/D' 

If  the  sodium  lines  are  to  coincide,  0=0",  or  approximately 

sin  (0-(*-j'))-sin*'  =  sin  0-cos  0  .  (i—i')—sini'=\/Dr 

or  on  eliminating  sin  0 

sin  i  —  sin  *'  —  (i  —  i')  cos  0  =  =  —  -= 

which  is  nearly 

,  ,  *-*'  _  X 

£>-£>'    £2(i-cos0) 

In  case  of  the  Wallace  grating  below 
£=1.75X10-*  X 


Thus  if  the  inclosed  angle  i—i'  between  the  plates  is  i  degree,  or  0.0175 
radian,  D—  7?'  =  5.3Xio-7,  about  0.3  per  cent  of  D  and  equivalent  to  about 
43  lines  to  the  inch.  With  adequate  facilities  for  measuring  i,  this  method 
may  be  useful  for  comparing  gratings,  not  too  different,  in  terms  of  a  normal 
or  standard,  practically,  since  the  finite  equations  may  also  be  expanded. 
In  a  similar  way  the  slight  adjustments  of  the  longitudinal  axes  of  the  two 
spectra  may  be  made  by  rotating  one  grating  around  a  horizontal  axis  ;  but 
this  correction  is  less  easily  specified.  Finally,  one  should  bear  in  mind  that 
with  film  gratings  there  is  liable  to  be  an  angle  i-4'  between  the  adjusted 
plates.  Fortunately  this  has  very  little  bearing  on  the  method  below. 

The  range  of  displacement  of  grating  within  which  the  fringes  may  be 
used  with  an  ordinary  small  telescope  extends  from  contact  of  the  two  gratings 
to  a  distance  of  e>=  2  to  3  cm.  beyond. 

In  figure  31,  which  is  a  plan  of  the  essential  planes  of  the  apparatus,  G,  G' 
being  the  ruled  faces  of  the  gratings  in  parallel,  7,  7',  7",  three  impinging 
rays  of  white  light  diffracted  into  D,  Df  ,  the  points  a,  b,  c,  a  ',  a",  b  are  in  the 
same  phase,  so  that  the  path-difference  of  the  rays  from  b  at  g  and  /  is  easily 
computed.  If  the  single  ray  7  is  diffracted  into  D  and  D'  or  7  and  7'  into  D, 
7  and  I"  into  D',  I'  and  7"  into  D  and  D',  the  equations  for  these  fringes 
should  be  (if  AP  is  the  path-difference), 


AP  =  e(i-cos  0)=e(i-Vi-X2/£2) 

where  D  is  the  grating  space,  e  the  distance  apart,  and  X  the  wave-length. 
Thus  the  micrometer  value  of  a  fringe  for  a  color  X  should  be,  under  normal 
incidence, 

~i-cos  0=  I-A/I_xyz?2 
For  two  colors  X  and  X' 

n\  =  e(i-cos  &)=eM  (n+n')\'  =  e(i-cos  tf}=eMe 


REVERSED   AND   NON-REVERSED   SPECTRA. 


49 


if  n'  is  the  number  of  fringes  between  X  and  X'.    Thus 

M'X-MX' 


(3) 


n  =  e- 


XX' 


or  the  number  of  fringes  increases  as  e  is  greater. 

Equation  (2)  does  not,  as  a  rule,  reproduce  the  phenomenon  very  well. 
Since  the  grating  space  D  of  the  two  gratings  is  rarely  quite  the  same,  the 
air-plate  inclosed,  in  case  of  apparent  coincidence  of  the  sodium  lines,  is 
slightly  wedge-shaped,  as  in  figure  32.  Hence  the  two  diffractions  take  place 
at  incidences  o°  and  a°,  respectively,  and  the  corresponding  angles  of  diffrac- 
tion will  be  0  and  0'.  If  we  consider  the  two  corresponding  rays  I  and  I", 
diffracted  at  the  first  and  second  face,  respectively,  and  coinciding  at  c  in  the 
latter,  the  points  a,  b,  and  a'  (bar  normal  to  ac),  are  in  the  same  phase,  and 
we  may  compute  the  phase-difference  at  the  coincident  points  at  c.  Since 
the  distance  be  is 

,  _  cos  a  cos  0 
cos     0  —  oi) 


32 


the  path-difference  is 

whence 
(4) 


33 


1      34 


e  cos  a  cos0 
-  r-n  -  c 
cos  (B-a 


.. 
—  cos  0) 


Xcos(0-a) 


cos  0  cos  a  •  (i  —  cos  0) 


which  changes  into  equation  (2)  when  a  =  o  and  w=i.  Fortunately  this 
correction  is,  as  a  rule,  small.  In  case  of  the  Wallace  gratings  (D  =  i .75  X  icr4 
cm.),  for  instance,  if  X=  58.93  X  iQ-8,  then  0=  19°  40'  or  de=  i.oi  X  iQ-3;  whereas 
if  a  =  5°,  then  te=  1.04X10-*;  if  a=io°,  then  fo?=  1.07X10-*,  etc. 

If  the  incidence  is  at  an  angle  i  and  the  plates  are  parallel,  figure  33,  the 
inquiry  leads  in  the  same  way  to  an  equation  of  more  serious  import.  If  the 
gratings  G  and  G'  are  at  a  distance  e  apart  and  the  incident  rays  are  I  and  /', 
the  points  a,  b,  c  are  in  the  same  phase.  Hence  the  two  rays  leaving  d  and 
diffracted  along  D  correspond  to  a  path-difference 


(5) 
whence 


cost 


v(i-cos(0-*)) 


X  cos  i 


i  — cos  (0  —  t) 


50 


THE   INTERFEROMETRY   OF 


Table  i  and  fig.  35  show  the  variation  of  fringes  with  the  angle  of  incidence  t, 
equation  (5).  Hence  if  the  angle  of  incidence  is  changed  from  —5°  to  +  5°, 
de  increases  to  nearly  3  times  its  first  value.  This,  therefore,  accounts  for  the 
large  discrepancies  of  8e  found  in  the  successive  data  below.  To  secure  in- 
creased sensitiveness  and  to  make  the  apparatus  less  sensitive  to  slight  changes 
of  i,  this  angle  should  be  about  25°,  in  which 
case  8e  is  about  three  wave-lengths  per  fringe.  But 
normal  incidence  is  frequently  more  convenient. 
Finally,  in  figure  34,  if  the  angle  of  incidence  is  i 
and  the  two  faces  G  and  G'  make  an  angle  a  with 
each  other  and  are  initially  at  a  distance  e  apart, 
changing  successively  to  e'  and  e",  the  points  o, 
b,  c,  being  in  the  same  phase,  the  two  rays  D  and 
D1  leaving  at  d,  at  an  angle  B—i,  will  have  a  path- 
difference  at  d  equal  to 
cos  0  cos  a 


whence 
(6) 


e : 77; \li — ^ua  \u  —  tyi 

cos  ^  cos  (Q— ay 

_         \cosicos  (6— a) 

cos  a  cos  6  (i  —  cos  (0— •&')) 
TABLE  i . — Wallace  gratings.    D=  io-*X  i  .75. 


i 

itfXSe 

f 

I03X5« 

+  19040' 

at 

-5° 

0.643 

+  15° 
+10° 

16.740 
4.090 

—  10° 

-15° 

0-443 
0.321 

+5° 

1.840 

-20° 

0.240 

±0° 

I.OI2 

-25° 

0.185 

This  equation  reproduces  the  preceding  equation  (5)  if  a  =  o  and  the  origi- 
nal equation  (2)  if  a  =  i  =  o.  It  shows  that  a  discrepancy  or  angle  between 
the  plates  is  of  minor  importance.  Hence  the  change  of  this  angle  may  be 
used  to  bring  the  sodium  lines  in  coincidence  when  the  gratings  differ  slightly 
in  their  grating  constants  D.  On  the  other  hand,  changes  of  incidence  *  are 
of  extreme  importance. 

Experiments  made  with  the  film  grating  showed  that  equation  (2)  not  only 
fits  very  badly,  but  that  8e  per  fringe  is  a  fluctuating  quantity.  Table  2 
gives  some  results  obtained  by  measuring  the  successive  values  obtained  for 
teXio8,  corresponding  to  10  fringes.  Fringes  were  distinctly  seen  within 
3  cm.  of  displacement  by  an  ordinary  telescope. 

TABLE  2. 
Gratings  15,050  lines  to  inch;  computed  values  io35c=o.94  cm.  per  fringe. 

First  sample:  Second  sample: 

io»«e=i.22  cm.  10^=0.98  cm. 


1.07 
1.16 


i. 06 
i. ii 


(normal  incidenc 


(oblique  incidence) 


REVERSED   AND   NON-REVERSED   SPECTRA.  51 

TABLE  2. — Continued 

Wallace  gratings  14,050  lines  to  inch;  computed  value  io33e=i.oi  cm.  per  fringe. 
io35c=i.22  cm.  (large  fringes);  io»Sc=i.35  (different  *') 
1-24  1-32 

1-25  1-30 

1-23  1.35 

1.23        (small  fringes)  1.33 

1-23  1.33 

1.32 
1-34 

The  reason  for  lack  of  accord  is  given  in  equations  (5)  and  (6)  and  table 
i.  Any  wedge  effect  of  the  glass  plate  is  probably  negligible.  To  show  that 
the  irregularity  of  the  above  results  is  to  be  sought  in  the  accidental  varia- 
tions of  the  angle  of  incidence  i  at  both  gratings,  the  rough  experiments  in 
table  3  suffice. 

TABLE  3. 

*,  negative  (less  than  10°),]         &Xio3=o.77  cm. 
Ocular  drawn  in,  [•  .78 

focus  changing.  J  .74 

*==to;  ocular  set  for  }  SeX io3=  1.18  cm. 

principal  focal  plane.    Na  lines 
in  field  and  coincident, 


*,  positive  (less  than  lo°),l  SeX  ios=  1.69  cm. 

ocular  drawn  out,  /  1  .66 

Thus,  as  equation  (6)  implies,  small  variations  of  i  produce  relatively  large 
variations  of  de,  and  if  *  passes  continuously  through  zero,  from  negative  to 
positive  incidence,  de  increases  continually  and  may  easily  be  more  than 
doubled.  If  the  phenomenon  is  in  focal  planes  in  front  of  the  principal  plane 
(ocular  in),  8e  is  small,  and  vice  versa.  Moreover,  this  enormous  discrepancy 
is  quite  as  marked  for  thin  glass  (2  mm.)  as  for  thick  glass  plates  (8  mm). 
Again,  the  rather  stiff  screw  of  the  micrometer,  which  twisted  the  whole 
apparatus  slightly,  was  sufficient  to  introduce  irregularity.  Placing  the  tele- 
scope close  to  the  grating  or  far  off  made  no  difference.  Hence  the  position  of 
the  optical  center  of  the  objective  does  not  affect  the  result. 

An  additional  result  was  obtained  by  placing  a  plate  of  glass  between  the 
two  gratings  G  and  Gf.  The  effect  was  an  unexpected  enlargement  of  fringes, 
increasing  with  the  thickness  of  the  glass  plate  (0.6  cm.  or  more).  The  reason 
for  this  is  given  by  equation  (3),  in  paragraph  2,  for  the  number  of  fringes 
n'  between  two  colors  X  and  X', 


XV 

where  M=  i-cos0,  M'=  i-cos0'.  Since  n'  is  a  number,  the  glass  plate  can  be 
effective  only  in  changing  6  and  0'.  As  both  are  diminished  by  refraction,  the 
cosines  are  increased  and  i-cos  0,  i-cos  0'  are  both  decreased.  Hence  n'  is 
decreased  or  the  number  of  fringes  is  decreased,  and  their  distance  apart  is 
thus  larger. 

It  is  obvious  that  when  the  sodium  lines  are  not  superposed  the  fringes 
can  not  He  at  infinity,  but  are  found  in  a  special  focal  plane,  depending  on 


52  THE   INTERFEROMETRY  OF 

the  character  of  coincidence;  i.e.,  whether  the  rays  are  convergent  or  diver- 
gent. Finally,  a  slight  rotation  of  the  slit  around  the  axis  of  the  collimator 
rotates  the  fringes  in  the  opposite  direction  to  the  sodium  lines,  and  it  is 
rather  surprising  that  so  much  rotation  of  slit  (10°  or  20°)  is  permissible 
without  fatally  blurring  the  image.  The  slightest  rotation  of  one  grating 
relatively  to  the  other  destroys  the  fringes. 

Naturally,  the  colored  fringes  vanish  when  the  slit  is  widened  or  when  it  is 
removed.  To  give  them  sharpness,  moreover,  the  beam  passing  through  the 
grating  must  be  narrow  laterally.  It  is  possible  to  see  these  colored  fringes 
with  the  naked  eye;  but  the  transverse  and  longitudinal  axes  must  in  this 
case  be  slightly  thrown  out  of  adjustment,  so  that  the  fringes  are  no  longer 
visible  in  the  telescope.  To  the  eye  they  form  a  somewhat  fan-shaped  set  of 
colored  fringes;  i.e.,  narrower  below  than  above.  Neither  are  the  lines  quite 
straight.  If  the  collimating  lens  is  removed,  a  slit  about  o.i  cm.  wide  across 
a  white  flame  will  also  show  (to  the  telescope  or  to  the  eye)  fine,  strong  lines 
rotating  in  opposite  direction  to  the  slit,  according  as  the  transverse  and  longi- 
tudinal axes  are  differently  placed.  As  has  been  already  stated,  it  is  with 
the  latter  condition  that  the  focal  plane  in  which  the  fringes  lie  varies  enor- 
mously. 

Finally,  when  the  sodium  lines  are  superposed  but  the  longitudinal  axis  of 
the  spectra  not  quite  so,  a  second  class  of  fringes  appear,  which,  however, 
are  always  more  or  less  blurred.  They  rotate  with  great  rapidity  over  180° 
when  one  grating  rotates  over  a  small  angle  relatively  to  the  other  and  the 
angle  between  the  longitudinal  axes  of  spectra  passes  through  zero.  In  the 
latter  position  the  regular  fringes  appear  in  full  strength  in  the  principal 
focus.  To  see  the  secondary  interferences,  the  ocular  must  be  drawn  inward 
(toward  the  grating),  and  these  fringes  increase  in  size  with  the  displacement 
of  the  ocular  away  from  its  position  when  regarding  the  principal  focal  plane. 
This  secondary  set  of  fringes  is  always  accompanied  by  another  very  faint 
set,  nearly  normal  to  them  and  apparently  quivering.  The  quiver  may  be 
due  to  parallax  and  the  motion  of  the  eye.  These  are  probably  the  vestiges 
of  the  regular  set  of  fringes,  out  of  adjustment. 

21.  Homogeneous  light.  Wide  slit.  Transverse  axes  coincident. — If  there 
is  no  color-difference,  fringes  of  the  same  kind  will  nevertheless  be  seen  in  the 
telescope,  on  widening  the  slit  indefinitely.  Path-difference  is  here  due  to 
differences  of  obliquity  in  the  interfering  rays.  As  in  the  preceding  case, 
accurate  adjustments  of  the  longitudinal  and  transverse  axes  (in  case  of 
sodium,  DI  and  D\  or  D2  and  D'z  coincide  horizontally  and  vertically)  of 
the  homogeneous  color-field  are  essential  if  strong  fringes  are  to  appear  in 
the  principal  focus.  These  fringes  are,  as  a  rule,  well  marked,  and  widening 
the  slit  merely  increases  the  width  of  the  channeled,  homogeneous  field  of 
view.  If,  owing  to  slight  differences  of  grating  space,  the  sodium  lines  are 
not  quite  superposed  automatically,  this  may  be  corrected  by  rotating  either 
grating,  or  else  the  apparatus  as  a  whole,  until  the  fringes  are  strongest.  The 


REVERSED   AND   NON-REVERSED   SPECTRA. 


53 


fringes  may  be  made  to  vanish  under  inverse  conditions.  Table  4  shows  their 
close  relation  to  the  preceding  colored  set,  so  far  as  motion  of  the  micrometer 
is  concerned. 

TABLE  4. 

Ives  grating.    15,050  lines  to  inch,  computed  Se=  io-3Xo.94  cm.  per  fringe. 
lo3Se=o.72  cm.  (large  fringes)     10*86=0.83  cm.  (small  fringes) 


.80 
•77 


.80 


Wallace  grating.    Wide  slit.    Coincident  Na  lines.    Fringes  in  principal  focus, 

very  clear  and  strong. 
lo33e=  1.16  cm.  io3de=l.o6  cm. 

small  f"ij  J'JJ 

fringes  Y*J| 

Wallace  grating.    Wide  slit.    Non-coincident  Na  lines. 

io35e=  1.34  cm.  io3Se=  1.38  cm. 

(small  fringes)    1.28  (large  fringes)  1.35 

The  fringes  decrease  in  size  as  e  increases  and  exhibit  the  same  irregularity 
of  8e  values,  due,  no  doubt,  to  the  same  causes  (equation  6).  Moreover,  8e 
is  here  below  the  normally  computed  value,  supposing  the  angle  i  to  be  negli- 
gible. In  fact,  figure  36  shows  the  optical  center  of  the  collimator  C;  so  that 


37 


36 


Ca  and  Ca'  are  the  axes  of  parallel  pencils,  diffracted  by  the  gratings  G  and  G' 
at  the  angles  6  for  Ca  and  e'  for  Ca'.  The  rays  are  subsequently  condensed 
at  F,  the  focus  of  the  telescope,  L  being  the  principal  plane  of  the  objective. 
The  general  path-difference  is  thus,  by  equation  (5),  e(i—cos  (0'-H"))/cos  i, 
which  distributes  the  fringes  from  right  to  left  with  variation  of  *. 


(small 
fringes) 


54  THE   INTERFEROMETRY  OF 

If  the  grating  G'  is  displaced  de  parallel  to  itself,  however,  the  path-difference 
will  again  be  increased  by  X  whenever 

_        X  cos  i 
1=  i -cos  (0'+*) 

Since  i  is  small,  this  equation  will  not  differ  appreciably  from  equation  (2), 
with  which  it  coincides  for  the  central  fringes. 

If  the  sodium  lines  are  not  superposed,  these  fringes  may  still  be  seen,  but 
they  are  not  in  the  principal  focal  plane  and  the  new  focal  plane  changes  con- 
tinually, as  the  fringes  grow  in  size.  Examples  are  given  in  table  4.  The 
large  values  of  be  show  that  i  was  not  actually  negligible.  Experiments 
similar  to  the  above,  bearing  on  the  reason  for  the  discrepancy  (equation  6), 
were  tried  with  the  thin  Wallace  gratings,  and  the  results  are  given  in  table  5. 

TABLE  5. — Thin  Wallace  grating. 

*  negative  (within  10°)  5eXio3=2.6o  cm.  5eXio3=2.4O  cm. 

Ocular  in,  2.34  2.45 

2-57 

*=±o,  normal  incidence,  8eX io'=  1.48  cm.  SeX io8=  1.37         ( (small 

Ocular  set  for  principal  1.50  1-37       '^fringes) 

focal  plane,  1.37  1.32         [(large 

1.19        \fringes) 

i  positive  (within  10°)      SeXio3=o.86  cm  SeX io*=  0.96 

Ocular  out,  .88  .86  cm 

•85 
.87 
.96  fringes) 

As  before,  the  effect  of  i  passing  from  negative  (through  zero)  to  positive 
values  is  enormous,  8e  increasing  nearly  threefold  for  a  change  of  *  estimated 
as  within  20.°  Here,  however,  the  drawn-out  ocular  (towards  the  observer) 
corresponds  to  the  small  values  of  8e,  whereas  above  the  reverse  was  the  case. 
This  depends  upon  which  of  the  spaces  D  or  D'  is  the  greater. 

22.  Homogeneous  light.    Fine  slit.    Transverse  axes  not  coincident. — To 

obtain  this  group  of  interferences,  the  two  sodium  lines  from  a  very  fine  slit 
are  thrown  slightly  out  of  coincidence;  i.e.,  by  not  more  than  the  DiD2  dis- 
tance. In  the  principal  focal  plane,  therefore,  these  doublets  are  seen  sharply, 
while  if  the  ocular  is  drawn  sufficiently  forward  or  rearward,  an  interesting  class 
of  fringes  soon  appears  which  resemble  Fresnel's  fringes  for  two  virtual  slits. 
These  fringes  may  be  seen,  however,  on  both  sides  of  the  focal  plane  and  in- 
crease in  size  with  the  distance  of  the  plane  of  observation  (focus  of  ocular) 
in  front  or  behind  the  principal  focal  plane.  In  figure  37  the  two  gratings, 
G  and  G',  are  struck  by  parallel  pencils  from  the  collimator  at  different  angles 
of  incidence  (o°  and  i0).  The  two  diffracted  pencils  of  parallel  rays  are 
caught  by  the  objective  L  of  the  telescope  and  condensed  at  the  principal 
foci,  F  and  F't  appearing  as  two  bright  yellow  lines.  In  front  and  behind  the 
plane  FF',  therefore,  are  two  regions  of  interference,  I  and  /',  throughout 
which  the  Fresnellian  phenomenon  may  be  seen  in  any  plane  parallel  to 
FF',  observed  by  the  ocular.  When  the  electric  arc  is  used  with  a  very  fine 


REVERSED   AND   NON-REVERSED   SPECTRA.  55 

slit,  these  sodium  fringes  often  appear  at  the  same  time  as  the  colored  fringes, 
and,  though  they  are  usually  of  different  sizes,  their  lateral  displacement 
with  a  change  of  distance  apart  of  the  gratings,  8e,  is  the  same.  The  fringes 
in  question  appear  alone  when  the  sodium  burner  is  used.  They  may  then 
(at  times)  be  observed  with  the  naked  eye,  with  or  without  a  lens,  and  they 
fail  to  appear  in  the  telescope  unless  the  objective  is  strengthened  by  an 
additional  lens.  They  are  always  vertical,  but  finer  in  proportion  as  the 
DiD2  and  D'iD'z  doublets  are  moved  farther  apart.  They  become  infinite 
in  size,  but  still  strong,  when  the  doublets  all  but  coincide,  showing  a  ten- 
dency to  become  sinuous  or  possibly  horizontal.  Rotation  of  either  grating 
G  around  an  axis  normal  to  itself  and  relative  to  the  other  produces  greatly 
enhanced  rotation  of  the  fringes,  as  in  all  the  above  cases,  but  they  soon 
become  blurred. 

Only  in  the  case  when  the  horizontal  axes  of  the  field  coincide  (parallel 
rulings,  etc.)  do  they  appear  strong.  When  the  angle  of  incidence  (or  non- 
coincidence)  is  increased  for  both  gratings,  the  size  of  the  fringes  increases; 
but  when  the  e  distance  is  increased  by  the  micrometer,  the  fringes  are  appar- 
ently constant  as  to  size.  However,  after  displacement  of  4  mm.  they  are 
liable  to  become  irregular  and  stringy,  though  still  moving.  A  fine  slit  is  not 
essential,  particularly  when  e  is  small.  They  vanish  gradually  when  the  slit 
is  too  wide.  If  a  telescope  with  a  strong  objective  is  used,  these  fringes  may 
be  seen,  retaining  their  constant  size  long  after  those  of  the  next  paragraph 
vanish.  Examples  of  data  are  given  in  table  6,  and  5e  is  too  low  in  value  as 
compared  with  the  computed  datum  for  i  =  o°.  With  the  Wallace  gratings, 
these  fringes  were  best  produced  by  the  aid  of  the  sodium  lines,  in  the  ordi- 
nary electric  arc,  simultaneously  with  the  colored  fringes  and  for  the  case  of 
a  very  fine  slit.  They  were  apparent  both  with  an  ocular  drawn  out  or  drawn 
in.  In  the  former  case  several  successive  groups  were  observed.  Beginning 
with  the  sharp  sodium  lines  in  principal  focus  (Dz  and  D'2  coincident),  a  slight 
displacement  of  the  ocular  outward  showed  the  first  group,  this  resembling  a 
grid  of  very  fine  striations.  Further  displacement  outward  produced  a  second 
set,  equally  clear  but  larger.  A  third  displacement  of  the  ocular  outward 
showed  the  third  set,  and  these  now  coincided  with,  and  moved  at,  the  same 
rate  as  the  colored  fringes  in  the  same  field.  Other  groups  could  not  be  found. 
No  doubt,  for  these  four  successive  steps  the  interference  grids  of  D\  and  D\, 
D2  and  D's  are  coincident  and  superposed,  until  they  finally  find  their  place  in 
the  colored  phenomenon. 

TABLE  6.  —  Ives  grating.    Homogeneous  light.    Fine  slit.    Sodium  lines  not  coincident. 
5eXio3=o.87  cm. 


23.  Homogeneous  light.  Slit  and  collimator  removed.  —  Fringes  similar 
to  those  seen  with  the  wide  slit  above  may  be  observed  to  better  advantage 
by  removing  the  slit  altogether.  The  sodium  flame  is  then  visible  as  a  whole; 
and  if  the  adjustments  are  perfected  it  is  intersected  with  strong,  vertical  black 


56 


THE   INTERFEROMETRY   OF 


lines,  visible  to  the  naked  eye  or  through  a  lens  or  a  suitably  strengthened 
telescope.  They  decrease  rapidly  with  increase  in  e,  but  vanish  to  the  eye 
before  the  preceding  set  in  paragraph  22.  The  sodium  lines  need  not  be  in 
adjustment,  but  the  longitudinal  axes  of  the  field  must  be,  as  usual.  If  diffuse 
white  light  is  present,  faint  colored  fringes  may  be  seen  at  the  same  time. 
If  the  collimator  only  partly  fills  the  field  of  view,  these  diffuse  light  fringes 
and  the  preceding  set  may  occur  together.  Both  rotate  markedly  for  slight 
rotation  of  either  grating  in  its  own  plane.  There  seems  to  be  a  double  peri- 
odicity in  the  yellow  field,  but  it  is  too  vague  to  be  discerned.  When  magni- 
fied with  a  lens,  they  admit  of  a  play  of  e  within  about  0.6  cm.  from  contact. 
When  the  sodium  lines  are  not  coincident,  the  focal  plane  continually  changes 
with  e.  Otherwise  it  remains  fixed. 
Some  data  are  given  in  table  7. 

TABLE  7. — Ives  grating.     Homogeneous  light.     Collimator  and  slit  removed.     Focus 
continually  changing. 

[  5eXio3=o.95  cm. 
Large  fringes;  ocular  out;  lens  on  \ 


Ocular  in,  lens  on 

Very  small  fringes,  lens  off  j 

Wallace  grating.    Sodium  lines  coincident. 
Principal  focal  plane 


i.oi 
i-<>4 
1.02 


6eX  io*=  1  .08  cm. 
1.18 


These  data  are  similar  to  the  above  and  subject  to  the  same  discrepancy 
whenever  slight  variation  of  the  angle  of  incidence  accidentally  occurs.    In 
figure  38  the  case  of  three  rays  from  a  given 
flame-point  F  is  shown  corresponding  to  the  ^ 

equation 

X  cos  * 
~~  i— cos  (0-H) 

when  i  passes  from  positive  to  negative  values. 
If  either  of  the  gratings  is  displaced  and  if  they 
are  parallel,  the  focal  plane  will  not  change; 
but  if  G  and  Gr  are  not  parallel,  the  focal  plane 
differs  from  the  principal  plane  and  now  moves 
with  the  grating. 

24.  Inferences.— The  above  data  show  that  the  equation  underlying  all  the 
interferences  observed  is  the  same.  The  interferences  themselves  may  result 
from  different  causes,  but  their  variation  in  consequence  of  the  motion  of  the 
grating,  8e,  is  due  to  one  and  the  same  cause.  This  is  best  seen  by  producing 
them  simultaneously  in  pairs.  As  a  means  of  finding  an  accurate  compari- 
son of  the  number  of  lines  per  centimeter  on  any  grating,  in  comparison 
with  those  on  the  given  grating,  the  method  used  in  paragraph  20  deserves 
consideration. 


REVERSED   AND    NON-REVERSED    SPECTRA.  57 

If  the  fringes  are  to  be  used  for  practical  purposes,  great  care  must  be  taken 
to  keep  the  angle  of  incidence  of  the  impinging  light  constant.  This  was  not 
done  in  the  present  paper,  where  the  purpose  is  merely  an  identification  of  the 
phenomena.  Moreover,  a  micrometer  with  the  screw  running  easily  is  essen- 
tial, as  otherwise  the  frame  is  liable  to  show  appreciable  twist  (change  of  inci- 
dence) during  displacement  of  the  fringes. 

The  fringes  are  not  of  the  sensitive  type,  but  they  admit  of  a  large  range  of 
displacement  and  are  therefore  adapted  to  special  purposes. 

With  regard  to  their  bearing  on  the  behavior  of  reversed  spectra,  for  the 
interpretation  of  which  the  present  experiments  were  undertaken,  it  is  obvious 
that  the  interferences  with  homogeneous  light  and  a  wide  slit  (paragraph  21), 
or  in  the  absence  of  a  slit  (paragraph  23),  are  of  analogous  origin  in  both 
cases.  It  makes  no  difference,  therefore,  whether  one  of  the  spectra  is  reversed 
or  not,  except,  perhaps,  that  in  the  former  case  (inversion),  the  coincidence 
of  longitudinal  and  transverse  axes  is  a  more  insistent  condition.  The  colored 
fringes  of  paragraph  20  obviously  can  not  be  produced  with  reversed  spectra. 
There  remain  the  fringes  with  the  fine  slit  and  homogeneous  light  (paragraph 
22)  ;  in  other  words,  the  occurrence  of  a  sort  of  generalized  Fresnellian  inter- 
ferences, within  the  telescope,  modified  by  causes  which  lie  outside  of  it.  Thus 
DI  and  D\  or  D2  and  D't  may  be  placed  sufficiently  close  together  to  pro- 
duce a  region  of  interference  before  and  behind  the  principal  plane  in  which 
the  sodium  lines  are  in  focus.  If  the  DiD'i  lines  are  o.oi  cm.  apart  and  the 
fringes  seen  likewise  at  o.oi  cm.  apart,  their  position,  measured  from  the 
principal  plane,  will  be  at 


or  less  than  2  cm.  The  ocular  would  then  have  to  be  displaced  forward  or 
rearward  by  this  amount.  But  there  are  two  sodium  doublets,  each  pair  of 
which  is  to  interfere.  Suppose  that  D2  and  D\  are  in  coincidence  so  that  the 


40 


scheme  is  DI\  DzD'i:  D'2,  as  in  figure  39,  where  o  is  the  principal  plane  of  the 
objective  and  DI  to  D'2  the  principal  focal  plane.  We  should  then  have  the 
separate  regions  of  interference  I  and  /'  and  the  combined  regions  I"  and  /'". 
When  the  breadth  of  the  latter  is  the  whole  number  of  fringes,  the  two  pat- 
terns clearly  merge  into  a  single  pattern.  The  experiments  show  several  of 
these  stages,  terminating  outermost  in  the  focal  plane  of  the  colored  fringes 
under  the  given  conditions.  Since  the  fringes  lie  on  hyperbolic  loci  the  problem 


58  THE   INTERFEROMETRY   OF 

itself  is  beyond  the  present  purposes;  but  it  appears  that  the  colored  fringes 
will  not  appear  until  the  corresponding  DI  and  D2  lines  are  shared  by  the 
whole  of  the  two  continuous  spectra. 

The  final  question  at  issue  is  the  bearing  of  the  present  Fresnel  phenome- 
non on  the  reversed  spectra.  If  in  figure  40,  5  and  s'  represent  the  traces  of 
two  reversed  spectra  in  the  principal  focal  plane,  superposed  throughout  their 
extent  (i.e.,  in  longitudinal  coincidence),  the  rays  a\a'iB,  through  the  line  of 
symmetry  a,  a',  are  at  once  in  a  condition  to  interfere  with  a  given  difference 
of  phase;  but  so  are  all  the  symmetrically  placed  pairs  of  colors,  c,  c',  b,  b',  of 
the  two  spectra  (the  distances  ccr,  bb',  being  arbitrary),  provided  the  corre- 
sponding rays  meet.  As  they  do  not  meet  in  the  principal  focus,  they  can 
interfere  only  outside  of  this — b  and  &'  at  B,  c  and  c'  at  C,  etc.  Similar  con- 
ditions must  hold  at  B'  and  C'  within  the  principal  focal  plane.  The  linear 
interference  is  thus  successively  transferred  to  different  pairs  of  wave-lengths. 
The  phenomena  of  this  paper  can  not,  therefore,  be  detected  in  case  of  reversed 
spectra,  because  in  the  principal  focal  plane  different  wave-lengths  are  every- 
where superposed,  except  at  the  narrow  strip  aa',  which  experiment  shows  to 
be  about  one-third  of  the  width  of  the  sodium  doublet,  in  apparent  size. 
Beyond  the  principal  focus  the  corresponding  conditions  in  turn  hold  for  the 
rays  at  B,  C,  etc.,  B',  C',  etc.  Hence  there  can  not  be  any  Fresnellian  inter- 
ferences (paragraph  22),  for  there  are  not  two  virtual  slits,  but  only  a  single 
one,  as  it  were,  and  the  interferences  are  laid  off  in  depth  along  the  normal 
C'C.  The  phenomenon  may,  in  fact,  be  detected  along  this  normal  for  2  or 
3  meters. 

25.  Rotation  of  colored  fringes.  Non-reversed  spectra. — When  the  slit  is 
oblique,  it  effectively  reproduces  the  wide  slit,  locally,  and  therefore  does  not 
destroy  the  colored  fringes.  At  every  elevation  in  the  field  the  slit  is  neces- 
sarily linear,  though  not  vertical.  In  figure  41,  let  the  heavy  lines,  H,  denote 
the  colored  fringes  for  a  fine  vertical  slit  and  white  light,  showing  nearly  the 
same  distance  apart,  throughout.  Let  the  light  lines,  L,  denote  the  fringes 
for  a  wide  vertical  slit  and  homogeneous  light,  X.  These  fringes  are  due  to 
the  successively  increased  or  decreased  obliquity  of  the  rays  in  the  horizontal 
plane.  Now  let  acb  be  the  image  of  the  oblique  slit  in  homogeneous  light.  It 
is  thus  merely  an  oblique  strip,  cut  from  the  area  of  light  lines  or  striations, 
as  it  were,  and  consists  of  an  alternation  of  black  and  bright  dot-like  vertical 
elements  in  correspondence  with  the  original  striated  field.  We  may  suppose 
ab  to  have  rotated  around  c,  so  that  the  vertical  through  c  is  its  position  on 
the  colored  field  (white  light  and  fine  vertical  slit). 

A  color,  X'  (near  the  one  X),  corresponding  to  the  field  of  the  lines  L  in  case 
of  a  wide  slit  and  homogeneous  light  X',  will  supply  nearly  the  same  grid,  so 
far  as  the  distance  apart  of  fringes  is  concerned.  But  the  grid  is  displaced 
laterally,  in  consequence  of  the  different  angle  of  diffraction,  6.  This  is  shown 
by  the  dotted  lines  D  in  figure  41,  the  effect  being  as  if  the  slit  had  been  dis- 
placed laterally.  If  the  wide  slit  for  homogeneous  light  X'  is  now  narrowed  and 


REVERSED   AND   NON-REVERSED   SPECTRA. 


59 


inclined  as  before,  an  alternation  of  bright  and  dark  elements  will  appear  in 
the  image  of  the  slit  ed,  corresponding  to  X'.  If  we  suppose  that  for  white 
light  and  the  fine  vertical  slit  the  position  of  the  fringe  (X')  was  at  cf,  we  may 
again  regard  c'  as  an  axis  of  rotation.  To  find  the  fringes  such  as  ff,  it  is  then 
only  necessary  to  connect  corresponding  black  elements  on  ab  and  ed.  Their 
inclination  is  thus  opposite  to  ab  and  ed,  or  they  have  rotated  in  a  direction 
opposite  to  that  of  the  slit.  If,  for  instance,  the  slit  image  ab  or  ed  is  gradu- 
ally moved  back  to  the  vertical,  the  points  g  and  h  will  move  with  great 
rapidity  and  in  both  directions  toward  infinity  and  the  fringes  //  and  /'/' 
become  vertical  lines  through  c  and  c',  respectively. 

It  is  interesting  to  inquire  into  the  frequency  of  fringes,  n,  when  the  angle 
of  diffraction,  6,  is  changed.  From  the  original  equation  e=n\/(i-cos  6), 
since  d\/d6=D  cos  6,  the  rate  of  change 

dn     e         i  e 


dd 


D  i-f-cos  6     D+V  £>2-X2 


where  e  is  the  distance  apart  of  films  and  D  the  grating  space.  Since  cos  6 
varies  but  slowly  with  6  and  is  additionally  augmented  by  i,  dn/dd  is  nearly 
constant  and  about  equal  to  e/2D. 


41 


/^ 


42 


The  fringes  and  slit  images  are  thus  given  by  the  two  sides  of  the  parallelo- 
gram cgc'h  for  the  two  colors  X  and  X'.  The  diagonal  cc'  represents  dd;  the 
diagonal  gh  has  no  signification.  On  the  other  hand,  the  normal  distances 
apart,  D'  and  D",  of//  and/'/  and  ab  and  ed  are  both  important. 

If  D'  and  D"  are  the  normal  distances  apart  of  the  fringes  and  the  slit 
images,  respectively,  B  and  Bf,  the  two  diagonals  of  the  rectangle  cgc'h,  modi- 
fied for  convenience  in  figure  42, 

£>' =£>"(cos  <f,+\/B'*/D"2  -  i  sin  <p) 

which  may  be  obtained  from  the  two  small  triangles  below  c' .  If  B=D", 
D'=D"  cos  <p;  and  if  <p  =  o,  D'—D'r=dd,  remembering  that  c  and  c'  lie  on  two 
consecutive  colored  fringes  obtained  with  white  light  and  a  fine  slit.  If  the 
slit  images  and  fringes  are  symmetrical,  each  is  at  an  internal  angle,  90-^/2, 
to  the  longitudinal  axis  of  the  spectrum. 


60 


THE   INTERFEROMETRY  OF 


But  these  equations,  though  useful  elsewhere,  have  very  little  immediate 
value  here,  because  the  experimental  variables,  figure  41,  are  B',  the  distance 
between  two  consecutive  colored  fringes  and  b"  and  b'  the  corresponding  dis- 
tance between  the  fringes  in  case  of  homogeneous  light  in  each  case  X,  X' ;  and 
the  angle  y',  which  indicates  the  inclination  of  the  slit.  Thus  B'b'b"  are 
given  by  computation  and  y'  is  specified  at  pleasure.  Obviously,  if  parallelo- 
grams are  to  be  obtained,  figure  41,  b'  =  b",  appreciably.  This  is  the  case  in 
experiment.  Hence  if  we  evaluate  the  height  in  the  triangle  cgc'  for  each 
angle  it  follows  easily  that 

tan  y' 


If  B'  =  b',  x'  =  go°  for  all  values  of  y';  i.e.,  the  fringes  remain  vertical.  If 
B'  is  equal  to  2&,  x'=yf,  the  fringes  and  slit  are  symmetrically  equiangular 
with  the  longitudinal  axis  of  the  spectrum.  This  is  nearly  the  case  in  figure 
41  and  frequently  occurs  in  experiment.  If  b'  differs  from  b",  the  fringes  would 
not  be  straight.  This  also  occurs,  particularly  when  the  thickness  e  of  the 
air-film  is  very  small. 

26.  Final  treatment  of  reversed  spectra.  Hypothetical  case. — To  obtain  an 
insight  into  the  cause  of  the  interferometer  fringes  as  obtained  with  reversed 
spectra  and  two  gratings,  it  is  convenient  to  represent  both  gratings,  figure 
43,  GG  and  G*G',  as  transmitting, 
and  suppose  both  diffracted  beams, 
ID'  and  ID",  subsequently  com- 
bined in  view  of  the  principal  plane 
PP  of  an  objective  or  a  lens.  It  is 
clear  that  this  simplified  device  can 
apply  only  for  homogeneous  light. 
In  the  case  of  white  light,  the  opaque 
mirrors  M  and  N  (of  the  interfer- 
ometer, above)  return  a  divergent 
colored  beam  or  spectrum,  so  that 
only  for  a  single  color  can  the  second 
incidence  be  the  same  as  the  first. 
Again,  if  the  constants  of  the  two 
gratings  are  different,  it  is  the  func- 
tion of  these  mirrors  to  change  the 
incidence  at  the  second  grating  correspondingly,  so  that  for  homogeneous 
light  the  rays  issue  in  parallel.  Finally,  no  reference  to  the  lateral  displace- 
ment OG"  and  OG'  of  rays  need  be  made  because,  as  more  fully  shown  in 
the  next  paragraph,  this  is  eliminated  by  the  theory  of  diffraction. 

The  motion  of  the  opaque  mirrors  M  and  N  (above),  on  a  micrometer, 
merely  shortens  the  air-paths  GG'  or  GG"  in  its  own  direction,  and  conse- 
quently the  same  fringe  reappears  for  an  effective  displacement  of  half  a  wave- 
length, as  in  all  interferometers. 


71' 


REVERSED   AND   NON-REVERSED   SPECTRA.  61 

The  case  of  a  single  grating,  moreover,  is  given  if  the  planes  of  the  grating 
GG  and  G'G"  and  their  lines  are  rigorously  parallel,  the  planes  OG'  and  G"0 
being  coplanar.  To  represent  the  interferences  of  the  two  independent  gratings 
and  with  homogeneous  light  for  the  case  of  oblique  incidence,  it  is  necessary 
to  suppose  the  grating  G'G"  cut  in  two  halves  at  0,  parallel  to  the  rulings, 
and  to  displace  the  parts  OG'  or  OG"  separately,  normally  to  themselves. 
Figure  43  shows  that  for  normal  incidence  i  =  o,  the  displacement  per  fringe, 
8e,  would  be 

X 

de  =  — 

i— cos  6 

or  the  fringes  are  similar  to  the  coarse  set  of  the  present  chapter. 

If  the  rays  impinge  at  an  angle  i,  figures  43  and  46,  they  will  be  parallel  after 
the  two  diffractions  are  completed;  for  it  is  obvious  that  the  corresponding 
angles  of  incidence  and  diffraction  are  merely  exchanged  at  the  two  gratings. 
Hence  the  homogeneous  rays  /',  impinging  at  an  angle  i,  leave  the  grating  at 
D'i  and  D"\  in  parallel,  at  an  angle  of  diffraction  i,  and  the  rays  unite  into  a 
bright  image  of  the  slit.  If,  however,  OG'  be  displaced  to  0\G'\,  parallel  to 
itself,  as  in  figure  44,  the  paths  intercepted  are 

— .and .cos  (0— i) 

cos  i        cos  i 

and  the  path-difference  per  fringe,  therefore, 

X  cos  i 


which  reduces  to  the  preceding  equation  if  «  =  o.  Hence  a  series  of  inter- 
ference fringes  of  the  color  X  must  appear  in  the  principal  focus  of  the  tele- 
scope or  lens,  on  either  side  of  *  =  o.  The  theory  of  diffraction  again  annuls 
the  apparent  path-difference  between  GG  and  G'G". 


44  45 


As  to  the  number  of  fringes,  n,  between  any  two  angles  of  incidence  i  and 
i' ',  it  appears  that  for  homogeneous  light  of  wave-length  X, 


_  g/i 

n  ~  X  \ 


_  i-cos 
cos  *  co 


62  THE   INTERFEROMETRY   OF 

where  e  is  the  distance  apart  of  the  two  parallel  halves  of  the  grating  G"0, 
OG'.  Hence  n  vanishes  with  e,  or  the  fringes  become  infinitely  large.  Lateral 
displacements  are  here  without  signification,  as  stated  above. 

If  the  grating  G'  is  rotated  over  an  angle  <p,  figure  43,  and  e=b<p  where  zb 
is  half  the  virtual  distance  apart  at  the  grating  G'  of  the  two  corresponding 
rays  impinging  upon  it  (Chapter  II,  fig.  26),  the  rotation  of  the  grating  per 
fringe  is  thus 

X        cos  * 


or  n  (above)  passes  through  zero  as  <p  or  b  decreases  from  positive  to  negative 
values.  If  b  is  considered  variable  there  is  a  wedge-effect  superposed  on  the 
interferences. 

It  is  this  passage  of  n  through  zero  that  is  accompanied  by  the  rotation  of 
the  fringes,  as  above  observed. 

In  case  of  two  independent  gratings,  GG  and  G'G"  (G'G"  to  be  treated  as 
consisting  of  identical  halves,  OG'  and  G"0),  nearly  in  parallel,  fringes  may 
be  modified  by  rotating  G'G"  around  the  three  cardinal  axes  passing  through 
the  point  of  symmetry  0.  The  rotation  of  G'G"  around  an  axis  O  normal  to 
the  diagram  is  equivalent  to  the  fore-and-aft  motion  of  G'G"  when  mirrors 
are  used  (fig.  26,  Chap.  II).  The  rotation  around  OT  in  the  diagram  and  nor- 
mal to  the  face  of  the  grating  requires  adjustment  at  the  mirrors  around  a 
horizontal  axis  to  bring  the  spectra  again  into  coincidence.  This  is  equiva- 
lent to  rotation  around  G"OG'.  Both  produce  enlargement,  and  rotation  of 
fringes  is  already  explained. 

Let  the  grating  G'G"  be  rotated  over  an  angle  <p  into  the  position  g'g",  figure 
45.  Then  the  angle  of  incidence  at  the  second  grating,  6,  on  one  side  is 
increased  to  6"=e+<p  and  on  the  other  decreased  to  6f=d—<f>.  In  such  a 
case  the  diffracted  rays  are  no  longer  parallel.  If  B'  and  B"  are  two  angles  of 
diffraction  on  the  right  and  on  the  left, 

sin  0'+sin  (0-^)=sin  (0+<p)  -sin  0"=X/L> 
whence 

sin  B  ''-{-sin  B'  =  2  sin  <p  cos  6 

or  if  B  is  the  mean  value  of  B'  and  B" 

8  =  <pcos  6,  nearly. 
Similarly,  since  sin  6=\/D,  for  i  =  o, 

sin  0'-sin  B"  =  2\  (i-cos  <p)/D 

Hence  only  so  long  as  <f>  is  very  small,  are  the  rays  appreciably  parallel  on 
rotating  G'G"  around  O  normal  to  the  diagram;  but  this  is  usually  the  case, 
as  <p  =  o  is  aimed  at,  and  fringes  are  thus  seen  in  the  principal  focus. 

To  the  same  degree  of  approximation  is  it  clear  that  on  rotating  the  grating 
into  a  position  such  as  og"  the  rays  emerge  parallel  to  IT,  figure  43. 

The  next  question  at  issue  is  the  rotation  of  fringes  with  fore-and-aft  mo- 
tion, or  rotation  around  an  axis  0  normal  to  the  diagram,  as  shown  in  figure  26, 


REVERSED   AND   NON-REVERSED   SPECTRA.  63 

Chapter  II.  In  other  words,  when  e,  the  virtual  distance  apart,  is  zero,  since 
ncce/\,  the  fringes  are  infinitely  large  horizontally.  The  collimator,  how- 
ever, furnishes  a  pencil  of  rays  which  are  parallel  in  a  horizontal  sectional 
plane  only.  They  are  not  collimated  or  parallel  in  the  vertical  plane  (parallel 
to  the  length  of  the  slit) .  Hence  when  the  fringes  are  reduced  to  a  single  one 
of  infinite  size  horizontally,  this  is  not  the  case  vertically;  i.e.,  from  top  to 
bottom  of  the  spectrum  the  path-difference  still  regularly  varies.  The  adjust- 
ment around  an  axis  through  0,  G'OG",  normal  to  the  rulings,  is  still  out- 
standing. It  does  not  seem  worth  while  to  enter  the  subject  further  because 
much  of  the  rotational  phenomenon  will  depend  upon  whether  the  axes  used 
are.  in  fact,  truly  vertical  or  parallel  to  the  slit.  In  my  apparatus  this  was 
not  quite  guaranteed,  and  the  quantitative  results  obtained  may  therefore  be 
due  to  mixed  causes.  Also,  a  rotation  around  an  axis  normal  to  0  always 
requires  an  adjustment  for  superposition  of  the  longitudinal  axes  of  the  spectra, 
and  this  introduces  path-difference. 

Finally,  the  case  of  figure  21,  Chapter  II,  or  the  rotation  around  an  axis 
parallel  to  IT  in  the  present  figure  43,  is  to  be  considered.  This  has  already 
been  given  in  terms  of  colored  fringes  (white  light),  but  it  occurs  here  for 
homogeneous  light,  in  which  case  the  above  explanation  is  not  applicable. 
Seen  in  the  principal  focal  plane  with  telescope  and  wide  slit,  the  non-reversed 
spectra  would  require  careful  adjustment  of  longitudinal  and  transverse  axes; 
otherwise  they  vanish.  Nothing  will  rotate  them. 

Figure  43  shows  that  if  G'G"  is  rotated  about  IT,  the  effect  is  merely  to 
destroy  the  fringes,  since  the  coincidence  of  the  longitudinal  axes  of  the  spectra 
is  here  destroyed.  No  effect  is  produced  so  far  as  path-difference  is  concerned. 
To  restore  the  fringes,  therefore,  either  of  the  opaque  mirrors  M  or  N  of  the 
apparatus  must  be  rotated  on  a  horizontal  axis  until  the  two  spectra  are  again 
longitudinally  superposed.  It  is  this  motion  that  modifies  the  path-difference 
of  rays  in  a  vertical  plane.  In  other  words,  when  the  fringes  corresponding 
to  any  virtual  distance  apart,  e  =  b  <?,  of  the  halves  of  the  grating  G'G", 
have  been  installed,  the  rays  as  a  whole  may  still  be  rotated  at  pleasure 
around  a  horizontal  axis.  In  this  way  a  change  in  the  number  of  fringes  inter- 
sected by  a  vertical  line  through  the  spectrum  is  produced.  The  number  of 
intersections  will  clearly  depend  on  the  obliquity  of  the  rays  (axes  of  vertical 
pencils),  and  will  be  a  minimum  when  the  center  of  the  field  of  view  corre- 
sponds to  an  axis  of  rays  normal  to  the  grating  G'G".  In  other  words,  the 
vertical  maximum  in  figure  22  occurs  under  conditions  of  complete  symmetry 
of  rays  in  the  vertical  plane.  If,  therefore,  e  or  the  virtual  distance  apart 
of  the  half  gratings,  G"0  and  OG',  is  also  zero,  the  field  will  show  the  same 
illumination  throughout. 

In  conclusion,  therefore,  to  completely  represent  the  behavior  of  fringes,  it 
will  be  sufficient  and  necessary  to  consider  that  either  grating,  G'G"  for 
instance,  is  capable  of  rotation,  not  only  around  a  vertical  axis  through  0, 
but  also  through  a  horizontal  axis  through  0  parallel  to  the  grating.  The 
last  case  has  been  directly  tested  above,  Chapter  II,  §  16.  But  a  rotation 


64  THE   INTERFEROMETRY  OF 

around  these  two  axes  is  equivalent  to  a  rotation  around  a  single  oblique 
axis,  and  the  fringes  will  therefore  in  general  be  arranged  obliquely  and 
parallel  to  the  oblique  axis. 

Thus  if  <pv  and  <ph  are  the  angles  of  rotation  of  the  grating  (always  small) 
around  a  vertical  and  a  horizontal  axis,  respectively,  and  if  xr  is  the  angle  of 
the  interference  fringes  with  the  horizontal  edge  or  axis  of  the  spectrum 


<Ph 

so  that  if  <f>v  =  o,  x'  =  o;  if  <ph  =  o,  x'  =  go°.  This  recalls  the  result  obtained 
above  for  the  interferences  of  two  coarse  grids.  In  other  words,  for  a  rotation 
of  grating  around  a  vertical  axis  (parallel  to  slit)  the  fringes  of  maximum  size 
will  be  horizontal  (Chapter  II,  fig.  21),  because  the  adjustment  around  the 
horizontal  axis  remains  outstanding  and  the  residual  fringes  (large  or  small) 
are  therefore  parallel  to  it.  For  a  rotation  of  grating  around  a  horizontal 
axis,  the  fringes  of  maximum  size  will  be  vertical  (Chapter  II,  fig.  22),  for  the 
vertical  adjustment  is  left  incomplete.  When  both  adjustments  are  made,  a 
single  fringe  fills  the  whole  infinite  field,  and  this  result  follows  automatically 
if  but  a  single  grating  is  used  to  produce  the  fringes,  as  in  the  original  method 
(Chapter  I). 

To  deduce  equations  it  is  convenient  to  regard  both  gratings  as  trans- 
mitting and  to  suppose  one  of  them  to  be  cut  into  independent  but  par- 
allel halves,  either  by  a  plane  through  its  middle  point  and  parallel  to  the 
rulings  (vertical  axis  of  rotation),  or  by  a  plane  through  the  same  point  and 
normal  to  the  rulings  (horizontal  axis  of  rotation).  The  parallel  halves  of 
the  grating  are  then  displaced  along  the  normal,  e,  to  both. 

27.  Case  of  reflecting  grating.  Homogeneous  light.  —  The  results  exhibited 
in  figure  43  for  transmitting  gratings  are  shown  in  figures  47  and  48  for  the 
combination  of  one  transmitting  grating  G  and  one  reflecting  grating  G'  (the 
adjustment  used  in  Chapter  II),  for  which  the  direct  path-lengths  of  rays 
were  computed  (cf.  figs.  23  and  24,  Chapter  II).  The  path-differences 
obtained  were  inadmissible.  It  is  now  necessary  to  completely  modify  the 
demonstration. 

In  figure  47  the  rays  are  shown  for  the  case  of  complete  symmetry  of  all 
parts,  gratings  at  G  and  G'  vertical  and  parallel,  opaque  mirrors  at  M\  and 
Ni,  telescope  or  lens  at  T.  The  incident  ray  I  at  normal  incidence  is  diffracted 
and  reflected  into  Y,  X,  T,  and  Y',  X',  T,  respectively;  the  incident  ray  I'  at 
an  angle  of  incidence  di  into  Yi,  Xi,  etc.,  and  Y\,  X\,  etc.,  respectively;  both 
at  a  mean  angle  of  diffraction  dd  (nearly)  to  the  right,  corresponding  to  di. 

The  angles  of  diffraction  (di=o]  are  61,  and  02;  the  double  angles  of  reflec- 
tion, therefore,  5=  02-  01(  on  both  sides;  the  double  angles  of  the  grating  G' 
with  the  mirrors  Mj.  and  Nlt  symmetrically,  <r=  0x+02. 

The  normal  from  the  point  of  incidence  at  G  and  at  G',  N,  and  «  makes 
angles  5/2  with  Y  and  X,  respectively,  on  both  sides.  The  method  of  treat- 


REVERSED   AND   NON-REVERSED   SPECTRA. 


65 


ment  will  consist  in  reflecting  G'  in  MI  and  NI,  producing  the  planes  G\  and 
G\  (virtual  images),  and  then  rotating  Mx  and  G\  180°  around  IT  (axis  of 
symmetry)  into  coincidence  with  Ni  and  G'2  (interference).  Then  the  rays 
prolonged  into  a  and  ft  coincide  with  the  rays  prolonged  into  a'  and  8'  and 
the  (virtual)  diffracted  rays  Tl  and  T2  become  T\  and  T'z.  The  ray  on  the 
left,  prolonged  into  e,  is  diffracted  into  Ts.  Then  the  interferences  will  all  be 
given  by  discussing  the  left  half  of  this  diagram,  which  is  amplified  in  figure  48. 


47 


Since  the  distance  GG',  figure  47,  is  very  large,  the  rays  are  nearly  parallel. 
Thus  the  arc  8'j,  with  its  center  at  G,  is  practically  a  plane  wave-front, 
perpendicular  to  the  rays  in  8',  ft',  y,  and  the  diffracted  rays  T',  T'2>  and  T'3 
are  also  practically  parallel.  Hence  in  the  case  of  symmetry  and  coincidence 
of  MI  and  Ni  the  points  8',  ft',  7,  8',  a',  and  e  are  in  the  same  phase  (diffrac- 
tion). In  other  words,  there  is  no  path-difference  between  Y-\-X  and  Y'-\-X', 
whether  the  angle  of  incidence  is  zero  or  not  (Yi+Xi  and  Y'i+X'i).  The 
whole  field  in  the  telescope  must  therefore  show  the  same  illumination  (homo- 
geneous light,  wide  slit)  between  the  maximum  brightness  and  complete 
darkness.  Interference  fringes  can  occur  only  when  the  opaque  mirror,  MI, 
is  displaced  parallel  to  itself  out  of  the  symmetrical  position.  If  MI  and  Ni  are 
symmetrical,  as  in  figure  47,  the  displacement  of  G',  fore  and  aft,  parallel  to 
itself,  is  without  influence. 

This  reduces  the  whole  discussion  to  the  normal  displacements  of  the  sys- 
tem G',  Mi,  NI,  given  in  figure  48.  Let  the  mirror  Mi  be  displaced  over  a 
normal  distance  em  to  the  position  M3)  Ni  remaining  in  place.  Then  the 
image  of  G'  will  be  at  G\,  at  a  perpendicular  distance,  e,  from  its  original  posi- 


66  THE   INTERFEROMETRY   OF 

tion  G'i.  The  path-difference  so  introduced,  since  a  and  b  (ab  normal  to  the 
ray  V2  impinging  on  Ms  at  c  and  reflected  to  &)  are  in  the  same  phase,  is 

26 m  COS  5/2 

and  the  displacement  per  fringe1  will  be 

5*m  =  — X— 

2  COS  5/2 

which  is  nearly  equal  to  X/2,  as  in  most  interferometers,  remembering  that 
em  and  dem  refer  to  the  displacement  of  the  mirror  M\.  Two  interfering  rays 
will  be  coincident  at  b. 

In  the  next  place  e  and  de  may  be  reduced  from  the  corresponding  displace- 
ments em  and  8em  of  the  mirror  M\.  In  figure  48  the  figure  Jdbe  is  approxi- 
mately a  parallelogram  with  the  acute  angles  5/2.  Hence,  since  02=  ( 5+00/2 

2em  cos<r/2=e 
as  is  also  otherwise  evident.    Thus  per  fringe,  if  the  length  £g  =  c 

X  =  5ecos  02+5csin  02 
since  5c  =  2  8em  sin  <r/2 . 

If  G'  is  displaced  parallel  to  itself,  de  will  not  be  modified,  since  each  virtual 
image  G'\  and  G's  moves  in  parallel,  in  the  same  direction,  by  the  same  amount. 
If  then  the  grating  G'  is  rotated  around  an  axis  at  G',  perpendicular  to  the 
diagram,  figure  47,  over  a  small  angle,  <p,  the  result  (apart  from  the  super- 
posed rotational  effect)  is  equivalent  to  a  displacement  of  the  mirrors  M\  and 
Ni  in  opposite  directions,  producing  a  virtual  distance  apart  e  and  the  cor- 
responding interference  fringes.  In  other  words,  the  rotational  effects  may  be 
explained  here  in  the  same  way  as  in  the  preceding  paragraph. 

The  angle  2dd,  within  which  the  interference  rays  lie,  per  fringe,  is  sub- 
tended by  dem,  and  this  may  be  put  roughly  (N  =  162  cm.,  normal  distance) 

2d6=  (2dem  sin  8/a)/N=  (X  tan  5/2)/AT 

This  angle  is  very  small,  scarcely  1 0^X3. 2  radians,  or  less  than  o.oi  second 
of  arc.  Hence  all  pencils  consist  of  practically  parallel  rays. 

An  important  result  is  the  angular  size  of  the  fringes;  i.e.,  if  em  and  X  are 
given 

_d62_         X        _  Dt  sin  02 
dn      em  sin  5/2     em  sin  5/2 

D2  being  the  grating  space. 

Thus  they  become  infinitely  large  when  em  passes  through  zero.  The  angu- 
lar size  is  independent  of  the  distance  between  the  gratings.  It  ought,  there- 
fore, to  be  easy  to  obtain  large  interference  fringes,  which  is  not  the  case. 
The  reason  probably  lies  in  this :  that  the  two  opaque  mirrors  are  not  quite 

iThe  differential  symbol  S  is  unfortunately  also  used  to  designate  the  double  angle  of 
reflection  5.  But  it  is  improbable  that  this  will  lead  to  confusion. 


REVERSED   AND   NON-REVERSED   SPECTRA. 


67 


symmetrical,  so  that  in  figure  47,  on  rotation  of  MI  18 S  on  GG',  the  trace  of 
Mi  crosses  Ni  at  an  angle.  If  d 0/dn  =  3.7X10-*,  the  distance  apart  of  the 
sodium  lines,  and  J92  =  173  X  io~*  cm., 

e=i.S  cm.,  about 

i.e.,  path-lengths  on  the  two  sides  should  differ  by  about  2  centimeters,  if  the 
mirrors  were  quite  symmetrical. 

28.  Non=symmetrical  positions.  Fore=and=aft  motion. — It  remains  to 
account  for  the  marked  effect  produced  on  displacing  the  grating  G'  in  a  direc- 
tion nearly  normal  to  it- 
self. If  the  displacement 
is  symmetrical,  or  even  if 
the  grating  and  mirrors 
are  reciprocally  non-sym- 
metrical (i-.e.,  the  former 
at  an  angle  <p  to  the  trans- 
verse line  of  symmetry 
gg',  the  latter  inclosing 
an  angle  a,  fig.  49),  no 
effect  results  from  the 
displacement  of  G',  if  the 
mirrors  MI  and  N\  are 
so  placed  that  the  vir- 
tual images  Gm  and  Gn 
are  parallel  and  the  dif- 
fracted rays,  therefore, 
also  parallel.  In  such  a 
case  Gm  and  Gn  are  dis-  \ 

placed  by  the  same  amount,  normally,  their  distance  apart  is  constant,  and 
the  intercepts  of  rays  equal. 

If,  however,  this  compensation  does  not  occur;  if  the  grating  G',  the  mirrors 
Ni  and  MI  make  angles  <p,  (r/2,  r/2,  respectively  (za=  r— <r),  with  the  trans- 
verse line  of  symmetry  gg',  the  fore-and-aft  motion  of  G'  is  more  effective  as 
the  angle  a  —  <p  is  greater.  The  diffracted  rays  are  then  no  longer  parallel, 
but  make  angles  of  incidence  at  the  second  grating,  0'2  for  the  NI  side  and 
02  for  the  Mi  side,  and  of  diffraction  *'  and  i,  respectively,  as  shown  in  figure 
49,  at  Tn  and  Tm.  The  following  relations  between  the  angles  are  apparent: 

If  at  the  first  grating  0i  =  0'i  and  a  is  the  angle  between  the  mirrors, 

20.— T  —  <T=  02  — 

The  images  are  at  an  angle  /3,  where 


49 


68  REVERSED   AND   NON-REVERSED    SPECTRA. 

If  G'G'  is  displaced  to  G\G\  over  a  normal  distance  e,  or  e/cos  <p  along  the 
line  of  symmetry  GT,  the  virtual  images  Gm  and  Gn  will  be  displaced  to  G'm 
and  G'n  over  the  same  normal  distance  e.  This  is  obvious,  since  the  quadri- 
laterals ab  and  a'b'  are  rhombuses  by  the  law  of  reflection,  and  hence  the 
perpendicular  distances  e  between  the  (equal)  sides  all  identical. 

If  Dz  is  the  grating  space  of  G', 


(1)  sin02+sint  =  X/.D2  sn 

or  if  *  and  i'  are  very  nearly  equal  and  both  small,  as  in  the  experiment, 

(2)  cos  8zd6=—  cos  idi 

Again,  in  case  of  the  displacement  e  of  G',  the  paths  are  shortened  at  Gm 
by  ej  cos  02,  at  Gn  by  e/  cos  0'2,  resulting  in  the  path-difference  AP,  or 

(3)  AP  =  <?(sec  02  -  sec  0'2) 

Since  02  and  0'2  are  nearly  the  same,  this  may  be  adequately  simplified  by 
differentiation.    Putting 

(4)  d0=02-0'2  =  2(a-<,2)  AP  =  2  (a  -<p)e  tan  02  sec  02 
Hence  per  fringe,  apart  from  sign, 

(5) 


.= 

2  (a  —  v?)  sin  02 
Thus,  if 

X  =  6Xicr5  a—  <p=i°  = 

then 


, 
2X0.0175X0.342 


=  0.004,  4  cm. 


per  fringe  for  each  degree  of  arc  of  non-symmetry,  a—  <p. 

The  effectiveness  of  the  fore-and-aft  motion,  according  to  this  equation, 
is  evidence  of  a  residual  angle,  a  —  <p,  of  non-symmetry.  This  is  not  improb- 
able, as  my  apparatus  was  an  improvised  construction,  lacking  mechanical 
refinement.  Further,  the  wedge  effect  due  to  a,  which  makes  em  variable, 
would  be  superposed  on  the  interferences,  and  hence  these  could  not  be  in- 
creased in  size  above  a  certain  maximum.  This  is  also  quite  in  accord  with 
observation. 

If  a  =  <p,  0  =  2(a  —  <p)=o  and  02=  0'2;  i.e.,  the  virtual  images  Gm  and  Gn  and 
the  diffracted  rays  are  parallel  and  8e  =  oo  .  In  other  words,  the  fore-and-aft 
motion  has  no  effect.  If  «  =  o,  &  =  2<p\  or  if  <p  =  o,  /3  =  2«.  In  either  case  8e 
is  finite,  and  fore-and-aft  motion  is  effective.  If  the  mirrors  and  grating  were 
rotated  in  counter-direction  so  that  <p  is  negative,  8e  will  depend  on  a-\-<p,  and 
the  fore-and-aft  effect  will  be  correspondingly  marked.  Moreover,  the  inter- 
ference will  not  in  general  appear  in  the  principal  focus,  but  usually  suffi- 
ciently near  it  for  adjustment. 

If  dea  is  the  actual  displacement  of  the  grating  G'  in  the  line  of  symmetry, 
dea  =  8e/  cos  <p,  so  that  the  angle  <f>  enters  equation  (5)  again,  but  only  to  a 
small  extent. 


CHAPTER  IV. 


THE  DISTANCE  BETWEEN  TWO  PARALLEL  TRANSPARENT  PLATES. 

29.  Introductory. — The  problem  of  finding  the  distance  separating  two 
parallel  glass  disks,  as  well  as  their  degree  of  parallelism,  is  frequently  one  of 
practical  importance.    Thus,  in  my  work  on  the  repulsion  of  two  such  disks, 
it  would  enter  fundamentally,  and  it  has  long  been  my  intention  to  repeat 
that  work  with  two  half-silvered  glass  disks,  for  comparison  with  the  case  of 
metallic  disks.    It  has  since  occurred  to  me  that  the  method  devised  by  my 
son,  Mr.  Maxwell  Barus,  and  myself*  would  probably  be  ideal  for  the  purpose, 
both  for  very  small  distances  (within  o.i  mm.)  as  well  as  for  distances  ten  or 
more  times  larger.     This  method  admits  of  use  of  the  film  grating,  and  there 
are  three  types  of  interferences  of  successive  orders  of  fineness,  the  first  virtu- 
ally involving  the  colors  of  thin  plates  (resolved  spectroscopically),  the  other 
two  being  dependent  on  diffraction.    To  measure  the  thickness  of  the  air-space 
it  would  be  necessary  to  count  the  number  of  fringes  between  two  definite 
Fraunhofer  lines  only,  supposing  the  constants  of  the  grating  to  be  given. 

30.  Apparatus. — The  apparatus  has  been  designed  for  transmitted  light, 
in  preference,  though  the  case  of  reflection  is  also  available. 


fc 

3 

aj= 

1     <S% 

^ 
Jl 

i= 

~f 

I 

^c^ 

y. 

[.,.., 

L 

ocl 

1 
c/H             50                   <^t 

V 


MM,  figure  50,  is  the  base  of  a  Fraunhofer  micrometer,  firmly  attached 
below  to  a  massive  tripod  (not  shown).  55  is  its  raised  slide,  and  E  the  head 
of  the  micrometer  screw,  reading  to  icr4  cm.  The  open  case  A  is  screwed  to 
the  slide  55  and  contains  the  glass  plate  H  half -silvered  on  the  right.  H  is 
attached  to  a  plate  of  brass,  on  the  plane-dot-slot  principle,  and  may  there- 
fore be  rotated  around  the  vertical  and  horizontal  axis  by  aid  of  a  rearward 
spring  mechanism  (not  shown)  and  the  adjustment  screws  a,  b,  b'  (the  last 
not  visible).  The  grating  G,  with  a  ruled  face  on  the  left,  is  similarly  carried 
by  the  open  rectangular  case  B,  screwed  down  to  the  base  M  of  the  micrometer. 
Thus  B  is  stationary,  while  A  moves.  Three  adjustment  screws,  c,  d,  d'  (d1 
not  shown),  and  a  spring  pulling  to  the  right  suffice  to  rotate  G  around  the 

*  C.  Barus  and  M.  Barus,  Carnegie  Inst.  Wash.  Pub.,  No.  149,  Part  I,  Chapters  II  and 
III.  1911. 


70  THE   INTERFEROMETRY  OF 

vertical  and  horizontal  axis.  The  thickness  of  the  efficient  air-film  is  thus  e- 
and  H  and  G  may  be  brought  to  touch  or  to  recede  from  each  other  several 
centimeters.  L  is  the  collimator  (slit  and  lens),  furnishing  intense  white  sun, 
light  or  arc  light,  and  the  beam,  after  traversing  the  system,  is  viewed  by 
the  telescope  T  (direct  beams,  fig.  51),  or  D  (diffracted  beams). 

The  plate  H  is  half-silvered,  but  the  grating  G  is  left  clear.  In  this  case, 
however,  only  the  fine  fringes  are  seen  strongly  on  transmission.  The  others 
appear  on  reflection  at  G,  preferably  in  the  second  order  of  spectra.  Fine 
fringes  are  not  well  reflected,  but  the  medium  and  coarse  fringes  are  very 
strong  and  clear,  and  the  first  observations  were  made  by  means  of  them. 

Thereafter  the  ruled  face  of  the  grating  was  half-silvered.  This  largely 
destroys  the  reflected  field,  D',  except  the  fine  fringes,  but  the  transmitted 
field  D  is  now  strong,  particularly  in  the  second  order  of  spectra,  for  all  the 
three  sets  of  fringes  in  question.  Mr.  Ives's  direct-vision  prism  grating 
shows  the  fine  fringes  well  in  the  direct  beam  T.  The  lines  are  always  rigor- 
ously straight,  so  far  as  they  can  be  observed;  i.e.,  it  is  impossible  to  bring 
H  and  G  rigorously  in  contact,  not  only  because  of  dust,  but  since  the  grating 
(at  least)  is  not  optical  plate.  The  fine  fringes  may  always  be  found  in  the 
principal  plane  of  a  telescope,  but  the  medium  and  coarse  fringes  usually  lie 
in  other  focal  planes  differing  from  each  other.  By  placing  the  ocular  it  is 
thus  possible  to  eliminate  any  of  the  interferences  or  to  show  a  single  set  in 
the  field  only. 

To  find  the  fringes,  the  direct  white-slit  images  are  made  to  coincide  through- 
out their  extent,  and  the  same  may  be  done  with  a  pair  of  spectrum  lines  in 
the  superposed  spectra.  The  proper  e  is  then  to  be  sought.  Owing  to  imper- 
fect plane  parallel  plates,  it  may  be  necessary  to  correct  this  by  the  adjustment 
screws  on  the  mirror  until  sharp,  strong  fringes  are  seen  in  the  corresponding 
focal  plane. 

31.  Equations.  —  The  equations  for  the  three  useful  interferences  in  ques- 
tion are  for  r  <  dm  and  a  similar  group  for  r  >  6m 


2)  n\  =  2en  cos  B'm 

(3  )  n\  =  2efjL  (cos  r  —  cos  ff'm) 

where  X  is  the  wave-length  of  the  color  used,  n  the  order  of  the  interference, 
e  the  thickness  of  the  sheet  to  be  measured,  and  /*  index  of  refraction,  if  i  is 
the  angle  of  incidence  of  the  white  light  on  the  grating,  r  the  angle  of  refrac- 
tion in  the  plate  (/*),  and  0'm  the  angle  of  diffraction  of  the  wth  order  of  spec- 
tra therein.  If  the  sheet  is  an  air-space,  these  equations  become  simplified, 
since  M  =  i  and  r  is  replaced  by  i,  d'm  by  8m,  the  angle  of  diffraction  in  air. 
Thus,  since  positive  values  are  in  question, 


n\  =  2e  cos  Bm 
(6)  n\  =  2<?(cos  i  —  cos  0m) 


REVERSED   AND   NON-REVERSED   SPECTRA. 


71 


In  the  present  apparatus  I  have  made  i  =  r  =  o,  a  more  convenient  plan 
of  testing  the  method,  though  not  necessary  and,  in  fact,  often  inconvenient 
in  practice.  The  equations  are,  finally, 

(7)  n\  =  2e 

(8)  n\  =  2e  cos  Om  =  2e-v/I_(mx/£>)2 

(9)  n\  =  2e(i  —  cos  0m)  =  2e(i—  A/! 


if  D  is  the  grating  space,  and  the  interference  in  question  is  due  to  the  grating 
spectrum  of  the  wth  order. 

The  meaning  of  the  equations  (7),  (8),  and  (9)  is  given  in  figure  52.  The 
case  of  equation  (7)  may  be  seen  in  the  direct  white  ray,  figure  52  a,  provided 
the  light  of  the  focussed  slit-image  is  resolved  by  direct-vision  spectroscope. 
For  this  purpose  Mr.  Ives's  grating  with  attached  direct  grating  prism  may 
conveniently  be  placed  in  front  of  the  telescope  T,  figure  50,  focussed  on  the 
slit.  After  adjustment  these  fringes  appear  strong.  Of  course,  H  and  G 


a 

b 

c 

\\  \\ 

OS 

\\   \V 

V  \ 

& 


52 


must  be  parallel  and  all  but  touch.  Under  the  same  conditions  the  fringes 
may  be  seen  laterally  in  any  order  of  spectrum,  as  in  figure  52  b.  Figure 
52  c  illustrates  equation  (8)  and  figure  52  b  equation  (9).  Figure  53,  finally, 
illustrates  the  general  case  of  incidence,  i. 

The  first  and  second  orders  of  spectra  are  alone  intense  enough  to  pro- 
duce marked  effects.  In  case  oii  —  Q,  a  double  diffraction  of  the  first  order, 
6'  reinforces  a  single  diffraction  of  the  second  order,  02,  since 

\/D  =  (sin  0'-sin  0)  =sin  02/2, 

(sin  e'-\/D)  =  (2\/D)/2  or  sin  e'  =  z\/D 

Probably  for  this  reason  they  are  visible.  The  general  case,  equations  (4), 
(5),  and  (6),  is  illustrated  in  figure  53,  the  rays  /,  I',  and  I"  being  incident, 
R  reflected,  and  D  diffracted.  The  retardations  are  ef  and  df,  respectively. 
If  the  diffractions  differ  by  a  whole  number  of  wave-lengths  the  total  diffrac- 
tion is  obtained.  One  would  be  tempted  to  resolve  the  case  by  aid  of  a  wave- 
front  ab,  in  which  case  the  equations  would  be  different;  but  they  do  not 
reproduce  the  phenomenon. 


72  THE   INTERFEROMETRY   OF 

32.  Method.— Suppose,  now,  two  Fraunhofer  lines,  X  and  X'  of  the  spectrum, 
are  selected  as  the  rays  between  which  interference  fringes  are  to  be  counted. 
Then,  in  case  of  equation  (7),  if  n'  is  the  number  of  interference  rings  between 
X  and  X', 

(I0)  n\ 

(n)  n 

(12)  2e 

In  order  to  measure  e,  therefore,  it  is  necessary  to  count  the  number  of  fringes 
n'  between  X  and  X',  and  e  varies  directly  as  n'. 

If  the  mean  D  and  magnesium  b  lines  be  taken  as  limiting  the  range,  io6X  = 
58.93  cm.,  io'X'  =  5i.75  cm.,  Ci=io-4X4-2S;  then 

n'=     i  io3e  =   0.21  cm. 

=   10  =   2.1 

=  100  =  21         etc. 

As  it  will  not  be  convenient  to  count  more  than  100  lines  ordinarily,  the 
method  is  thus  limited  to  air-spaces  below  0.2  mm.  and  becomes  more  avail- 
able as  the  film  is  thinner.  Of  course,  in  case  of  plates  which  contain  specks 
of  dust  or  lint,  or  are  not  optically  flat  on  their  surfaces,  it  is  extremely  diffi- 
cult to  get  e  down  below  0.002  cm.,  so  that  ten  fringes  between  D  and  b  would 
require  very  careful  preparation. 

If  equation  (8)  is  taken,  X  is  to  be  increased  to 


L  =  X/  cos  0m  =  X/Vi  -  (m\/DY 

where  m  is  the  order  of  the  grating  spectrum,  whose  rays  interfere.      Thus 
equations  (n)  and  (12)  now  become,  since  nL  =  (n+n')L'  =  2e 

(13)  n  =  n'L/(L-L'} 

(14)  2e  =  n'LL'/(L-L')=C2n' 

If  first  order  of  diffractions  are  in  question,  ra  =  i,  io6L  =  59.11,  io6L'  =  52.33, 
£•'2=10^X4-20.    Thus  for 

«=     i  io3e=     0.21  cm. 

10  02.1 

100  021. 

scarcely  differing  from  the  preceding  case,  so  that  one  would  not  know  in 
which  series  one  is  working. 

If  the  diffractions  occur  in  the  second  order,  m  =  2, 


62.56  io6L'2=  54.15  C"2=  10^X4.03 

thus  again  differing  but  slightly  from  the  above. 

If  we  inquire  into  the  condition  of  coincidence  and  opposition  of  these 
fringes,  the  following  results  appear:  Let  the  spectrum  distance  between 
the  G  and  6  line  be  taken  as  unity,  and  let  there  be  m  and  n,  first-order  fringes 
in  this  distance.  Then  in  ~±  is  the  difference  of  distance  per  fringe.  Let 


REVERSED   AND   NON-REVERSED    SPECTRA.  73 

x  be  the  number  of  long  fringes,  to  restore  the  original  coincident  phase;  i.e., 
let  x  longer  fringes  gain  one  long  fringe  on  the  x  shorter  fringes.    Then 


that  is,  x  fringes  constitute  a  new  period.    From  the  above  data 


It  follows  that  the  length  of  coincident  strips  is  subject  to 

or  C- 


where  C  is  the  new  constant.  This  would  place  the  fringes  beyond  the  coarse 
group  below,  but  naturally  C  is  enormously  dependent  on  small  errors  in  C\ 
and  C2.  - 

Finally,  if  equation  (9)  be  taken,  the  X  is  to  be  increased  to 


M  =  X/(i  —  cos  0m)=X/(i  —  V  i  — 
in  order  that  equations  similar  to  the  above  may  apply.    Thus 

n'M  MM'  , 

''    M'-M  '    M'-M 

In  the  diffractions  of  the  first  order  of  spectra  m=i  and  io3M  =  4.i5o, 

io3M'  =  4-747  £"'3  =  0.0330 

These  are  the  coarse  order  of  fringes,  so  that 

n=     i         20  =  0.033 

10  0.33 

100  3.3     ,  etc. 

Fringes  are  thus  still  strongly  available,  even  if  the  distance  apart  of  the 
plates  is  over  2  cm. 

If  the  diffractions  are  of  a  second  order  of  spectra,  m  =  2, 

io3M=i.oi6  ioW=i.i6s  C'"s=io-*X7.85 

These  fringes  are  therefore  of  intermediate  order,  since 

n=     i         20  =  0.0078  cm. 

10  0.0785 

100  0.785 

They  would  be  enhanced,  since  they  cooperate  with  the  double  diffractions 
of  the  first  order. 

33.  Observations  and  corrections.  Preliminary  work. — The  following 
work  was  done  merely  with  a  view  to  testing  the  equations  and  with  no 
attempt  at  accuracy.  The  grating  was  left  unsilvered,  so  that  the  ruled  sur- 


74 


THE    INTERFEROMETRY   OF 


faces  confronted  the  half-silvered  surface  on  ordinary  plate  glass.  Conse- 
quently, the  fine  fringes  were  observed  by  transmitted  light  behind,  and  the 
medium  and  coarse  fringes  by  reflected  light  in  front.  The  micrometer  was 
a  good  instrument  for  general  purposes,  but  hardly  equal  to  the  present  work, 
where  the  slightest  rocking  of  the  slide  introduces  annoyances. 

To  count  the  number  of  fringes  between  D  and  b,  since  the  fringes  were  not 
generally  seen  in  the  principal  focal  plane  of  the  telescope,  it  was  considered 
sufficient  to  rotate  the  cross-hair  into  an  oblique  position,  until  its  ends  ter- 
minated in  the  D  and  b  lines,  respectively,  and  then  to  count  the  number  of 
fringes  on  running  the  eye  down  the  wire  from  end  to  end.  When  there  are 
many  fringes,  25  to  50,  the  eye  is  apt  to  tire  before  reaching  its  destination, 
so  that  several  counts  must  be  made  and  the  mean  taken. 


0      •/ 


0      -01 


The  results  are  given  as  a  whole  in  figure  54,  where  the  distance  between 
plates,  measured  in  centimeters  on  the  micrometer,  beginning  at  an  approxi- 
mate zero,  is  laid  off  horizontally  and  the  number  of  fringes  vertically,  in  case 
of  each  of  the  three  series.  The  computed  line  e  =  Cn'/z  is  drawn  in  full  and 
the  observations  laid  off  with  regard  to  it.  The  zeros  do  not  quite  correspond, 
as  very  small  distances  here  are  significant.  With  the  fine  fringes  I  did  not 
spend  much  time,  as  they  are  virtually  colors  of  thin  plates  seen  by  diffrac- 
tion. The  chief  difficulty  with  these  small  distances  is  that  the  plates  touch 
and  a  complete  readjustment  is  necessary.  After  touching,  the  micrometer 
acts  like  a  forcing  screw  and  its  reading  is  too  low.  This  is  the  meaning  of 
the  data  in  the  curves  a  and  a',  the  latter  with  its  horizontal  scale  magnified 
ten  times.  The  object  of  this  series  is  chiefly  to  locate  the  position  of  the 
line  in  relation  to  the  other  lines. 


REVERSED   AND   NON-REVERSED   SPECTRA.  75 

The  observations  for  medium  and  fine  fringes  were  made  together,  so  that 
a  single  micrometer  reading  suffices.  Beginning  with  very  small  distances 
apart,  called  zero,  this  was  rapidly  increased  to  nearly  i  cm.  The  fine  fringes 
soon  vanished,  later  the  medium  fringes  vanished,  finally  (when  e  is  several 
centimeters)  the  coarse  fringes  also  vanish.  The  three  together,  therefore, 
cover  with  accuracy  a  relatively  enormous  range  of  displacement  for  measure- 
ments of  this  kind. 

The  curves  b  and  c  show  that  the  observations  are  not  completely  repro- 
duced by  the  line.  Mean  lines  drawn  through  the  observations  indicate  that 
the  zeros  do  not  correspond  sufficiently  for  the  two  lines  b  and  c  to  locate  the 
common  zero.  This  is  inevitable,  since  the  micrometer  begins  to  count  at  a 
small  distance  as  specified,  which  is  otherwise  arbitrary.  In  fact,  it  should 
be  noticed,  as  an  accessory  property  of  this  interferometry,  that  the  two  lines 
for  finding  the  zero  determine  the  absolute  reading  of  the  micrometer,  mutu- 
ally, and"  these  readings  are  here  0.22  mm.  too  large.  But  even  if  the  zeros 
were  horizontally  to  coincide,  the  observations  would  not  adequately  conform 
to  the  computed  lines.  All  that  can  be  affirmed  is  that  the  angle  between 
the  observed  and  computed  loci  is  about  the  same. 

The  main  reason  for  the  divergence  is  referable  to  the  fact  that  the  air-space 
is  not  quite  plane  parallel,  but  slightly  wedge-shaped,  so  that  the  effect  of  the 
angle  of  the  wedge  is  superposed  on  the  interferences.  Any  slight  unsteadi- 
ness of  the  micrometer  slide,  for  instance,  would  already  introduce  the  wedge 
discrepancy,  without  necessarily  interfering  with  the  sharpness  of  visibility, 
while  any  attempt  to  readjust  would  destroy  the  continuity  of  measurement. 
There  will  also  be  many  secondary  reasons  for  divergence,  as,  for  instance,  the 
three  separate  focal  planes  in  which  the  fringes  lie  and  the  fact  that  the  glass 
plates  which  limit  the  air-space  are  themselves  wedge-shaped;  other,  but 
fainter,  fringes  are  marching  through 
the  spectrum,  such  cases  as  coincidence 
and  opposition,  for  instance,  as  were  CA 

pointed  out  above,  etc.  But  the  ade- 
quate reason  for  the  discrepancies  in 
this  paper  is  the  incidental  change  of 
the  angle  of  incidence,  i. 

If  the  film  is  wedge-shaped,  very 
little  disturbance  results;  but  the  cor- 
rection to  the  second  order  of  small 
quantities  is  unfortunately  somewhat 
cumbersome.  Let  the  edge  of  the  wedge  of  air  be  vertical  and  subtend  a  small 
angle,  <f>,  figure  55,  between  the  two  faces  A  and  B.  Let  I  and  /'  be  the  two 
corresponding  rays  incident  at  the  angle  i  at  the  first  face  and  at  the  angle 
i-\-V  at  the  second  face,  n  and  ri  being  the  normals.  Let  e  and  e'  be  the  con- 
secutive thicknesses  of  the  air-plate,  taken  normal  to  B  for  convenience.  Then 
the  I  rays  R'  will  issue  at  the  A  face,  after  reflection,  at  an  angle  i+z<p,  and 
will  interfere  with  the  I'  rays  R,  if  the  objective  of  the  telescope  is  sufficiently 


76  THE   INTERFEROMETRY   OF 

large  to  converge  both  to  the  same  point  of  the  image,  spectroscopically 
resolved.    If  the  wave-front  ab  is  drawn  and  e'  prolonged,  it  follows  at  once 

that  ,_  /     ,    sin  (i+y)  sin  y 

n\  =  2e  cos  (*+f  J  e  -  e\ *  +  2 "        cos  * 

HenCC  /  s]n(t+^)sin^ 

If  i—o  and  ^>  very  small,  this  becomes 


If  in  the  first  equation  i  is  replaced  by  the  angle  of  diffraction  6,  the  equation 
for  the  diffracted  fringes,  as  far  as  <p2,  may  be  reduced  to 


n\  = 


-^(^-0-7  ^s  e)] 


so  that  <p  sin  0  is  the  chief  correction. 
Finally,  the  equation  for  the  coarse  fringes  becomes 


0)] 


with  a  similar  equation  for  the  medium  fringes. 

If  we  neglect  the  second  order  of  small  quantities  (<p2),  the  last  equation 
for  the  medium  fringes  may  be  put  in  another  form,  since 

2\/D  =  sin  6  and  X/L  =  i  -  cos  6  sin  6  =  ^(i  -  cos  0) 

whence 

nL  (n-t-n')L' 


i  -  2<pL/D  ~  i  -  2<pL'/D 
D  being  the  grating  space  and  X  the  wave-length.    Hence  if  n  be  eliminated, 


-nL-L'-(2<p/D}(LL'-LL'}-"L-L' 

In  other  words,  if  ?  is  smaU  so  that  ^  may  be  neglected,  the  relation  of  e  and 
n'  is  independent  of  <p\  or  a  slightly  wedge-shaped  air-film  will  show  the  same 
result  as  a  plane-parallel  film.  Experiments  made  by  turning  the  adjust- 
ment screws  seem  to  bear  this  out,  provided  the  mean  thickness  remains 
unchanged. 

To  give  the  whole  subject  further  study,  I  have  since  half-silvered  the 
grating  as  specified,  so  that  all  the  fringes  may  be  seen  by  transmitted  light, 
preferably  in  the  second  order,  since  there  is  an  abundance  of  light  available. 
The  apparatus  in  such  a  case  takes  a  good  shape  and  is  convenient  for  manip- 
ulation. But  these  details  will  have  to  be  given  at  some  other  time,  and  it 
is  the  chief  purpose  of  this  paper  to  exhibit  the  phenomenon  as  a  whole. 


REVERSED   AND   NON-REVERSED    SPECTRA. 


77 


In  conclusion,  I  may  recall  that  if  we  regard  100  fringes  between  the  D 
and  b  lines  as  still  available  for  counting  under  proper  facilities,  the  succes- 
sive ranges  of  measurements  will  be  roughly  as  follows : 


e=o.O2i  cm. 

5=0.392  cm. 

6=1.65  cm- 

Fine  fringes  

n'=ioo 

Medium  fringes  
Coarse  fringes  

»'=  54 
n'=     1.3 

n'=ioo 

n'=  23.8 

n'=ioo 

The  transition  from  fine  fringes  to  medium  is  a  little  abrupt.  Otherwise,  in 
cases  where  manual  interference  is  not  permissible,  all  thickness  of  air-films, 
from  a  fraction  of  a  wave-length  of  light  to  nearly  2  cm.,  may  be  adequately 
measured  in  this  way  to  advantage.  It  is  probable,  moreover,  that  it  would 
be  advisable  to  observe  the  fine  fringes  by  transmitted  light,  but  to  leave  the 
grating  (which  may  be  a  film  grating)  clear,  and  to  observe  the  medium  and 
coarse  fringes  by  reflected  light.  A  concave  mirror  and  lens  (reflecting  tele- 
scope) should  be  used  for  this  purpose,  as  this  will  put  the  observer  behind 
the  plates  in  all  cases. 


CHAPTER  V. 


INTERFEROMETERS  FOR  PARALLEL  AND  FOR  CROSSED  RAYS. 

34.  Introduction.  Methods. — To  exchange  the  component  beams  of  the 
interferometer,  to  mutually  replace  the  two  pencils  which  interfere,  is  not  an 
unusual  desideratum,  for  instance,  in  the  famous  experiment  of  Michelson 
and  Morley.  To  replace  two  pencils  of  component  rays,  traveling  more  or 
less  parallel  to  each  other,  by  pencils  moving  more  or  less  normal  to  each  other, 
or  to  be  able  to  operate  upon  pencils  of  corresponding  rays  (from  the  same 
source,  crossing  each  other  at  any  angle)  at  their  point  of  intersection,  may 
be  of  interest  in  a  variety  of  operations  to  which  the  interferometer  lends 
itself,  or  may  even  suggest  novel  experiments.  The  facility  with  which  this 
may  be  done,  or  at  least  partially  done,  with  the  above  types  of  spectrum 
interferometers,  particularly  when  homogeneous  light  is  used,  has  tempted 
me  to  investigate  a  number  of  cases. 

Let  us  begin  with  the  above  diagram- 
matic method,  using  two  transmitting 
gratings,  G  and  G',  figure  56,  with  the  same 
(or  in  general  with  different)  grating  con- 
stants. Let  L  be  the  incident  beam  of 
collimated  homogeneous  light,  m,  n,  m',  n', 
four  opaque  mirrors  on  vertical  and  hori- 
zontal axes  parallel  to  their  faces.  The 
ruled  faces  of  the  gratings  are  to  be  toward 
each  other.  Then  the  beams  Gm  and  Gn 
may  be  reflected  either  across  each  other, 
as  shown  at  mn'  and  nm',  thence  along  n'G' 
and  m'G',  and,  after  a  second  diffraction  at 
G',  unite  to  enter  the  telescope  at  T;  or 
they  may  be  reflected  along  m,  m',  and 
n,  n',  parallel  to  each  other,  and  thereafter  take  the  same  course.  In  the 
first  case  homogeneous  light  is  apparently  not  necessary.  It  will  be  seen 
that  the  path  of  the  rays  is  the  same,  except  for  the  branches  mn'  and  nm', 
and  mm'  and  nn't  respectively  normal  and  parallel  to  each  other;  moreover, 
that  the  rays  are  exchanged,  a  and  b  left  and  right  combining  at  G'  in  one  case, 
b  and  a  left  and  right  in  the  other.  The  rays  cross  at  c  in  free  space  and  are 
available  there  for  experiments.  Direct  light  is  to  be  screened  off.  The  ques- 
tion is  whether  the  mirrors  m  and  n,  m'  and  »',  can  be  adjusted  mechanically 
to  move  symmetrically  toward  each  other  on  a  vertical  axis  with  sufficient 
precision  to  guarantee  replacement.  This  is  a  matter  of  trial,  though  a 
successful  issue  is,  of  course,  problematical.  It  would  be  advantageous  to 
arrange  the  experiment  so  that  only  one  pair  of  mirrors— e.g.,  m  and  n— need 
78 


REVERSED   AND   NON-REVERSED   SPECTRA.  79 

be  moved,  whereas  the  others,  m'  and  n',  are  ends  of  the  same  rigid  plate. 
Gratings  of  different  constants  may  advantageously  contribute  to  this  end. 
Beyond  this,  the  paths  mn'  and  nm'  and  mm'  and  nn'  may  be  increased  to 
any  length,  either  directly  or  by  multiple  reflections  from  a  special  system. 
Many  other  modifications  are  suggested.  If  white  light  is  used,  the  phe- 
nomenon is  confined  to  a  narrow  strip  of  spectrum  and  the  fringes  must  be 
horizontal. 

As  I  did  not  have  two  ruled  transmitting  gratings  and  as  film  gratings 
seemed  unpromising  for  work  of  this  kind,  the  method  of  figure  57  represents 
a  simple  disposition  of  reflecting  gratings,  of  which  several  were  available. 
The  ruled  faces  of  the  gratings  G'  and  G  face  away  from  each  other. 


The  former  receives  the  collimated  pencil  of  homogeneous  light,  L,  and 
after  diffraction  the  partial  beams  pass  to  the  pair  of  opaque  mirrors  m  and 
n  (symmetrically  placed),  and  thence  by  reflection  to  a  similar  pair  of  mirrors, 
M  and  N.  From  here  the  pencils  reach  the  second  grating,  G',  where  each 
is  again  diffracted  into  the  common  ray  G'T,  entering  the  telescope  T.  The 
grating  G'  may  be  concave  with  the  lens  at  T  beyond  the  principal  focus. 
If  the  mirrors  M  and  N  are  symmetrically  rotated,  the  parallel  component 
pencils  Nn  and  Mm  may  be  replaced  by  the  pencils  Mn  and  Nm,  crossed  at 
any  angle.  Homogeneous  light  is  preferable.  Simultaneously  the  rays  are 
exchanged.  The  pencils,  Mm,  etc.,  may  be  of  any  length,  and  in  general  the 
remarks  in  the  preceding  paragraph  apply. 

A  more  flexible  design  also  suggests  itself,  with  four  fixed  mirrors,  m,  n, 
m',  n',  four  movable  mirrors,  M,  N,  M',  N',  rotating  symmetrically  around 
vertical  axes  parallel  to  the  faces  of  the  gratings  G  and  G',  these  being  parallel 
to  each  other,  as  in  figure  57.  On  rotating  M,  N,  M',  N',  the  rays  may  be 
exchanged.  Here  M  .  .  .  .  N'  should  be  a  near  system,  m  .  .  ,  .  n'  a  fixed 
and  far  system  of  mirrors.  Other  methods  will  presently  be  described. 

35.  Experiments.  Reflecting  gratings.  Parallel  rays. — The  experiments 
were  begun  with  the  apparatus  as  in  figure  57,  G  being  a  Michelson  grating 
and  G'  a  Rowland  grating,  each  with  somewhat  less  than  15,000  lines  to  the 
inch.  The  distance  of  G  from  the  mirrors  m  and  n  was  about  22  cm.,  of  G 
from  G'  about  60  cm.,  and  of  G'  to  the  focal  point  just  ahead  of  the  lens  (or 
the  line  of  mirrors  M  and  N)  about  90  cm.  The  latter  were  about  50  cm. 
apart.  In  the  absence  of  sunlight,  the  arc  lamp  was  used,  and  the  fringes  for 
reversed  spectra  were  found  without  great  difficulty.  It  was  also  easy  to 


80  THE   INTERFEROMETRY   OF 

erect  them  by  rotating  G'  on  an  axis  normal  to  its  face.  A  difficulty,  however, 
existed  in  retaining  the  fringes  with  a  flickering  arc.  It  will  be  seen  that  in 
this  case  the  line  LG  moves  over  a  small  angle  in  all  directions  with  the  bright 
spot  on  the  positive  carbon,  so  that  the  angle  of  incidence  is  varied,  and  with 
it  the  angle  of  diffraction  0  at  G.  All  this  is  magnified  by  reflection  from  the 
mirrors.  Moreover,  unless  the  collimator  lens  is  very  near  G,  the  illuminated 
part  or  bright  line  on  G  is  displaced  right  and  left.  Path-difference  between 
GnNG'  and  GmMG'  is  thus  modified.  If  the  faces  of  the  mirrors  are  not  all 
quite  in  a  vertical  plane  or  parallel  to  the  same  plane,  the  up-and-down  play 
of  the  arc  will  mar  the  longitudinal  coincidence  of  the  two  superposed  spectra, 
and  hence  the  interferences  will  vanish.  Thus  they  appear  and  disappear 
periodically,  depending  on  the  accidental  position  of  the  bright  spot  of  the 
arc;  and  if  this  annoyance  is  to  be  avoided,  sunlight  or  a  steady  light  must 
be  used.  The  phenomenon  and  the  spectra  were  not  nearly  so  bright  as 
when  observed  with  the  transmitting  grating,  a  result  probably  due  both 
to  the  additional  reflections  (particularly  those  at  the  grating)  and  to  the 
high  dispersion. 

In  other  respects  the  behavior  was  the  same  as  that  described  in  Chapters 
I  and  II,  though  the  strip  of  fringes  for  reversed  spectra  seemed  to  be  some- 
what broader,  probably  owing  to  the  increased  dispersion  and  hence  the  greater 
breadth  of  adequately  homogeneous  spectrum  light.  The  linear  phenome- 
non, moreover,  consisted  of  two  or  more  black  lines  alternating  with  bright, 
whereas  a  single  black  line  was  the  characteristic  feature  above.  When  dif- 
ferent strips  of  the  grating  G  are  used  (the  illumination  should  not  be  more 
then  0.5  cm.  wide),  considerable  fore-and-aft  displacement  at  the  mirror  M 
is  necessary. 

The  adjustment  for  crossed  rays  Mn  and  Nm,  figure  57,  is  subject  to  new 
conditions.  In  case  of  white  light  and  a  narrow  slit,  the  dispersion  produced 
by  G  is  at  least  partially  annulled  by  G'  instead  of  being  incremented;  for  the 
change  of  the  angle  of  incidence  here  compensates  the  changes  of  the  angle 
of  diffraction.  Thus  if  sin^—  sin  6v  =  \v/D  for  violet  and  sin*V—  sin  6r  = 
\r/D  for  red,  and  if  sin  iv  =  \v/D  and  sin  ir  =  \r/D,  then  sin  6  =sin  0r  =  o. 

A  sharp,  white  image  of  the  slit  may  thus  be  seen  for  the  reflection  from  each 
mirror  M  and  N,  or  the  images  may  be  colored  if  but  a  part  of  the  spectrum 
is  reflected  from  M  and  N.  The  system  of  two  gratings,  G  and  G',  tends  to 
become  achromatic.  It  would  seem  to  follow,  therefore,  that  in  general 
homogeneous  light  and  a  wide  slit  would  have  to  be  used,  but  this  introduces 
additional  annoyances,  inasmuch  as  the  transverse  axes  of  the  spectra  (sodium 
lines),  which  are  to  coincide,  are  not  visible,  but  must  be  replaced  inade- 
quately by  the  edges  of  the  slit.  The  experiment  is  thus  (particularly  in  view 
of  the  faint  illumination  seen  in  the  telescope)  difficult,  and  in  a  laboratory 
not  free  from  agitation,  or  in  the  absence  of  a  good  mercury  lamp  of  intense 
homogeneous  light,  it  did  not  seem  worth  while  to  spend  much  time  on  it. 
Moreover,  a  similar  investigation  will  presently  be  made  with  a  transmitting 
grating. 


REVERSED   AND   NON-REVERSED   SPECTRA.  81 

In  other  words,  in  case  of  the  rays  nM,  the  violet  is  incident  at  a  larger 
angle  at  G'  than  the  red,  and  but  one  color  (yellow)  can  be  diffracted  along 
G'T,  whereas  in  case  of  the  rays  mN  violet  is  incident  at  G'  at  a  smaller  angle 
than  red,  and  G'  may  thus  be  so  placed  that  all  rays  are  diffracted  along  G'T, 
supposing  the  two  gratings  to  be  nearly  identical  as  to  dispersion.  Figure 
58,  presently  to  be  described,  suggests  the  inclination  of  the  successive  verti- 
cal planes  in  figure  57. 

One  curious  result  deserves  special  mention.  Each  separate  spectrum 
(a  or  b,  fig.  57,  without  superposition)  shows  very  definite  coarse  stationary 
interferences;  i.e.,  the  usual  appearance  of  channeled  spectra.  The  cause  of 
this  long  remained  obscure  to  me,  but  will  be  explained  in  Chapter  VI.  The 
gratings  being  of  the  reflecting  type  and  the  mirrors  silvered  on  the  front  face, 
there  is  no  discernible  cause  for  interferences.  No  film  or  set  of  parallel  plates 
enters  into  the  experiments.  If  in  figure  57  the  grating  G'  is  reflected  at  M 
into  G'i,  and  this  image  reflected  in  m  into  G't,  the  phenomenon  may  be  treated 
as  if  the  gratings  were  transmitting  in  a  manner  shown  in  figure  58.  Here  the 
direction  of  the  traces  of  the  grating  G  and  G',  the  mirrors  m  and  M  only 
are  given,  together  with  the  direction  of  the  reflected  images  of  G'  in  M 
(G'i),  and  in  m  (G'z).  Then  the  violet  (v)  and  red  (r)  rays  from  G  impinge 
on  G'z  virtually  with  a  greater  angle  for  v  and  a  smaller  one  for  r,  as  already 
suggested.  An  enhanced  spectrum  must  be  produced  beyond  G'z.  This 
second  spectrum  is  channeled. 

36.  Experiments.  Transmitting  grating.  Parallel  rays. — The  chief  diffi- 
culty in  the  preceding  experiments  was  the  absence  of  sufficiently  intense 
homogeneous  light.  This  may  be  obviated  by  using  the  transmitting  grating. 
But  as  two  samples  were  not  available  (as  in  fig.  56),  the  simplified  method  of 
figure  59  was  tested,  where  but  a  single  grating  G  is  used.  Here  the  light  L 
from  collimator  and  slit  impinges  on  the  grating  G  and  is  diffracted  to  the 
opaque  mirrors  M  and  AT.  From  here  it  is  reflected  to  the  corresponding 
opaque  mirrors  m  and  n,  to  be  again  reflected  to  the  grating  G,  and  finally 
diffracted  along  the  line  GT.  The  interferences  are  observed  by  the  telescope 
at  T.  In  order  that  the  undeviated  white  beam  may  not  enter  the  telescope 
annoyingly,  the  diffraction  LG  takes  place  in  the  lower  half  of  the  grating  and 
the  mirrors  are  slightly  inclined  upward,  so  that  the  second  diffraction  GT 
may  occur  in  the  upper  half  of  the  grating.  To  obviate  glare  in  the  field,  the 
beam  LG  is  carried  to  the  grating  in  an  opaque  tube  and  all  undeviated  light 
is  suitably  screened  off.  The  distances  mn  to  G  and  G  to  MN  were  about  a 
meter  each. 

The  interferences  were  easily  found.  They  are  usually  at  an  angle  to  the 
vertical,  but  may  be  erected  by  rotating  the  grating  on  an  axis  normal  to  its 
face.  They  were  linear  and  exactly  like  the  cases  of  Chapter  I,  probably  in 
consequence  of  the  low  dispersion  of  the  grating  used.  Considerable  mag- 
nification at  the  telescope  is  thus  admissible. 

The  horizontal  fringes  traveling  up  or  down  are  available  for  interferometry, 


82 


THE   INTERFEROMETRY   OF 


and  the  independent  and  separated  component  beams  Mm  and  Nn  are  con- 
veniently accessible. 

The  experiments  with  homogeneous  light  (sodium  arc)  gave  perfectly 
regular  striations  covering  the  whole  of  the  wide  slit  image,  uniformly.  With 
glass  compensators  0.6  to  o.i  cm.  or  more  thick  on  both  sides,  the  striations 
became  somewhat  smaller,  as  was  to  be  anticipated.  Fringes  could  be  erected 
and  enlarged  by  rotating  the  grating  on  an  axis  normal  to  its  face  and  by 
other  corresponding  rotations.  The  fringes,  as  a  whole,  were  large  and 
splendid  and  suitable  for  general  purposes  in  interferometry. 

37.  Experiments.  Transmitting  grating.  Crossed  rays. — The  second  posi- 
tion of  this  apparatus  was  now  tested,  the  rays  passing  along  the  diagonal 
of  the  rectangle  (fig.  59)  and  crossing  at  G  in  the  grating.  The  interfering 
pencils  were  thus  GNGmG  and  GMGnG.  The  slit 
should  be  quite  wide.  Seen  in  the  telescope  at  T, 
therefore,  the  dispersion  is  reduced  in  virtue  of 
double  diffraction,  the  tendency  being  toward  white 
slit  images,  as  already  explained.  A  variety  of  very 
interesting  results  were  obtained  after  the  interfer- 
ences had  been  found.  The  outgoing  and  returning 
paths  are  coincident,  and  both  component  rays  pass 
through  the  grating  two  times,  the  ruled  face  being 
towards  the  telescope. 

The  adjustment  is  at  first  somewhat  difficult. 
Having  made  a  rough  setting  of  the  mirrors  as  to 
distance,  etc.,  by  the  aid  of  sunlight  or  arc  light,  so 
that  the  spectra  may  be  seen,  two  wide  slit  images 
will  appear  in  the  telescope  T,  but  they  will  usually 
be  differently  colored.  The  mirrors  m  and  n  are  then  to  be  rotated  around 
vertical  axes  (fine-screw  motion)  until  both  slit  images  are  identically  colored 
and  coincide.  After  this,  homogeneous  light  (sodium  arc)  must  be  used  and 
the  rotation  of  mirrors  on  the  vertical  and  horizontal  axes  repeated  until  both 
fields  are  identically  yellow  on  coincidence.  The  sharply  focussed  edges  of 
the  wide  slit  are  now  the  vertical  and  horizontal  guide-lines  for  adjustment. 
All  corresponding  lines  must  coincide  if  the  phenomenon  is  to  be  obtainable. 
Thereafter  the  micrometer  at  M,  actuating  the  mirror  fore-and-aft  parallel 
to  itself,  is  manipulated  till  the  fringes  appear. 

Two  types  of  interference  may  be  observed.  The  first  are  variations  of 
nearly  equidistant  fringe  patterns,  obtained  with  homogeneous  light  only 
and  covering  the  whole  wide  slit  image  on  good  adjustment.  They  would 
appear  equally  well  in  the  absence  of  the  slit.  The  second  type  is  obtained 
in  the  presence  of  white  light,  or  of  the  mixture  of  white  light  with  the  homo- 
geneous light.  It  is  a  linear  phenomenon,  identical  in  appearance  with  the  one 
described  in  Chapter  I,  though  occurring  here  in  the  case  of  a  wide  slit.  Both 
are  very  vivid,  and  the  latter  particularly,  when  at  its  best,  in  violent  tremor. 


REVERSED  AND   NON-REVERSED   SPECTRA.  83 

It  is  convenient  to  describe  the  homogeneous  fringes  first.  White  light 
must  be  absent,  the  wide  field  full  yellow,  and  the  longitudinal  and  side  edges 
of  the  two  slit  images  sharply  superposed.  When  the  fringes  appear  they  will 
usually  be  oblique;  but  they  may  be  made  vertical  by  rotating  the  grating 
on  an  axis  normal  to  its  face.  If  the  grating  is  in  the  symmetrical  position  of 
figure  59,  the  size  of  fringes  is  an  intermediate  minimum.  To  enlarge  them, 
curiously  enough,  the  grating  must  be  slightly  rotated,  either  way,  on  a  ver- 
tical axis.  The  fringes  then  pass  through  a  maximum  of  size  at  a  definite 
angle  on  either  side  of  the  minimum.  In  such  a  case  they  also  appear  rapidly 
to  become  irregular  and  their  perturbation  is  naturally  enhanced.  They  con- 
tain a  double  periodicity,  which  will  presently  be  carefully  examined. 

Fore-and-aft  motion  of  the  grating  has  no  effect.  In  displacing  the  mirror 
at  M  on  the  micrometer,  the  fringes  remain  visible  for  an  excursion  of  at  least 
0.7  cm.  In  fact,  in  case  of  a  strong  telescope  and  wide  slit  they  were  not  lost 
for  a  micrometer  displacement  of  over  i  cm.,  i.e.,  much  over  30,000  wave- 
lengths of  path-difference.  As  a  rule,  the  fringes  are  strong  only  in  part  of 
the  yellow  field,  and  in  such  a  case  the  center  of  intensity  moves  with  the 
displacement  of  M  across  the  slit  image,  to  disappear  at  the  edges,  as  in  the 
usual  cases  of  displacement  interferometry.  Slight  non-coincidence  of  the 
horizontal  edges  of  the  slit  images  slightly  rotates  the  fringes,  but  they  soon 
vanish  completely.  Slight  rotation  of  the  grating  around  the  vertical  axis 
distributes  the  fringes  more  evenly  over  the  field,  the  proper  setting  being 
determined  by  trial.  Displacement  by  aid  of  a  compensator  of  glass  gave  the 
usual  results. 

Later  I  returned  to  the  experiments  with  sodium  light  and  with  the  grating 
rotated  around  a  vertical  axis  to  the  right  or  left  and  out  of  the  symmetrical 
position  of  figure  59.  In  each  case  the  fringes  passed  through  maximum  size 
at  an  angle  of  asymmetry  of  about  5°  or  10°  from  a  normal  position.  Beyond 
or  below  this  they  diminish  in  size.  Naturally,  to  bring  the  fringes  to  the 
center  of  the  field,  the  micrometer  screw  at  M  or  AT  had  to  be  adjusted  for 
path-difference,  as  in  displacement  interferometry  generally. 

The  details  of  the  interference  patterns  obtained  were  in  astonishing  variety. 
Suppose  that  by  rotating  the  grating  around  an  axis  normal  to  its  face  the 
fringes  are  made  nearly  but  not  quite  vertical  at  the  beginning.  Then  on  rota- 
ting the  grating  around  a  vertical  axis  into  the  position  for  maximum  size 
just  specified,  the  standard  type  of  large  fringes  seen  are  of  the  appearance 
shown  in  figure  6oa.  In  other  words,  they  look  and  behave  like  independent, 
thick,  twisted  cords,  hung  side  by  side.  The  evolution  of  these  independent 
parallel  striations  of  fringes  may  be  detected  on  rotating  the  mirror  M  or  N 
around  a  vertical  axis,  thus  moving  one  slit  image  in  definite  amounts,  micro- 
metrically,  over  the  other,  horizontal  edges  remaining  superposed.  As  the 
one  slit  image  passes  in  this  way  across  the  other,  the  original  type,  figure  606, 
apparently  continuous,  breaks  up  and  enlarges  into  the  type  c  by  the  rotation 
of  its  parts.  Thus  the  successive  lengths  of  the  continuous  fringe  b  behave 
like  a  series  of  magnetic  needles,  each  rotating  on  its  own  pivot.  These  may 


84  THE   INTERFEROMETRY  OF 

again  correspond  and  appear  as  a  single  striated  field;  but  more  frequently 
the  form  figure  6oa  is  in  evidence,  though  sometimes  quite  irregular.  In  fact, 
there  are  many  variations  of  this  design.  Families  of  curves,  intersecting  each 
other  nearly  orthogonally,  may  even  appear. 

If  the  fringes  are  originally  quite  vertical,  there  seems  to  be  no  rotation, 
but  two  sets  of  vertical  fringes  apparently  pass  through  each  other  as  the 
mirror  M  is  rotated  micrometrically  on  a  vertical  axis.  These  fringes  at  inter- 
vals again  unite  into  an  apparently  simple  striation.  One  slit  image  may  be 
broader  than  the  other.  Fringes  of  different  sizes  then  appear,  so  long  as  the 
smaller  is  within  the  larger,  and  are  most  intense  when  the  vertical  edges  meet. 
In  general,  therefore,  the  interference  patterns  of  originally  nearly  vertical 
fringes  consist  of  a  succession  of  strands,  nearly  in  parallel,  which  behave  alike 
but  independently. 

61 


If  the  grating  is  rotated  on  an  axis  normal  to  its  face  until  the  fringes  are 
nearly  horizontal,  a  correlative  series  of  interesting  phenomena  may  be 
observed.  When  the  grating  is  normal  to  the  incident  pencil,  the  fringes  are 
usually  arranged  in  parallel  strands.  They  are  equidistant  in  each  strand ;  but 
these  strands  are  separated  by  a  narrow  band  of  even  color,  so  that  the  phe- 
nomenon looks  as  if  thick,  twisted,  yellow  cords  were  hanging  apart,  side  by 
side.  Usually  the  central  or  the  two  central  cords  are  more  intense,  and  there 
may  be  four  to  six  in  all,  filling  the  whole  of  the  wide-slit  image.  On  rotating 
the  mirror,  M  or  N,  micrometrically,  on  a  vertical  axis,  the  fringes  of  the 
strand  may  be  made  to  correspond,  so  as  to  fill  the  field  with  uniform  stria- 
tions  and  without  apparent  vertical  separation.  This  is  particularly  the  case 
when  the  fringes  are  very  fine. 

On  rotating  the  grating  to  the  right  or  to  the  left  about  20°,  on  the  vertical 
axis  from  the  symmetrical  position  of  figure  58,  the  fringes  reach  a  maximum 
of  size,  after  which  (on  further  rotation  to  about  30°)  they  diminish  indefi- 
nitely. These  maximum  cases  are  shown  in  figure  61,  a  and  b,  and  their  ap- 
pearance is  now  that  of  a  string  of  elongated  beads,  hung  vertically  and  equi- 
distant. On  rotating  N  about  a  vertical  axis,  slightly,  the  nodules  become 
quite  horizontal.  They  are  continually  in  motion,  up  and  down,  and  quiver 
about  the  horizontal  position  like  small  disturbed  magnetic  needles.  At  times 
the  field  appears  reticulated  (indicated  in  the  figure),  as  if  two  sets  of  nearly 
horizontal  fringes  intersected  at  a  small  angle.  It  is  now  difficult  to  obtain 


REVERSED   AND   NON-REVERSED    SPECTRA.  85 

continuous  striations  on  rotating  N,  but  the  whole  field  may  easily  be  filled 
with  nodules.     The  occurrence  of  two  maxima  is  probably  an  incidental 
result,  as  in  other  adjustments  but  a  single  one  appeared.    Naturally  the  rota- 
tion of  the  grating  or  of  the  mirrors  M  and  N  changes  the  path-difference  of 
the  pencils  crossing  within  it,  so  that  the  micrometer  screw  ////•*'*.  \\\> 
at  the  mirror  M  must  be  moved  in  compensation.    Thus  w     4  »    ^  w 
this  is  another  method  of  displacement  interferometry  and   ///,     £    »    ^  \\ 
the  usual  equation  suffices.  ^    m,    t  in!  y.' 

The  following  rough  experiments  were  made:    Placing  ,  ~ 

the  strong  fringes  in  the  center  of  the  field  (slit  image),  ^ 

the  reading  of  the  micrometer  was  taken.  Then  a  thick  glass  plate,  0  =  0.71 
cm.,  was  inserted  in  one  beam,  nearly  normally,  and  the  micrometer  displace- 
ment, A/V,  was  found  when  the  fringes  were  brought  back  to  the  center  of  the 
field  again.  The  results  were  (for  instance) 


0.393  cm. 
The  displacement  equation  is  (p  being  the  index  of  refraction  of  the  plate) 


where  the  correction  for  dispersion  may  be  put  2.B/X2  =  o.o26.  Hence  M  = 
1.50,  1.52,  as  was  anticipated.  On  using  white  light,  where  there  is  but  a 
single  strand,  a  cross-hair,  and  greater  care  as  to  the  normality  of  the  plate 
compensator,  etc.,  there  is  no  reason  why  results  of  precision  should  not  be 
obtained. 

38.  The  same.  The  linear  phenomenon.  —  The  occurrence  of  the  linear 
phenomenon  reciprocally  with  the  fringes  for  homogeneous  light  is  interesting. 
It  usually  appears  when  there  is  a  flash  of  the  arc  lamp,  i.e.,  a  displacement 
of  the  crater,  introducing  white  light  into  the  sodium  arc.  It  is  thus  undoubt- 
edly due  to  the  reversed  spectra  for  white  light  and  may,  in  fact,  be  produced 
by  using  the  white  arc  or  sunlight  in  place  of  the  sodium  arc.  When  the 
mirror  M  is  displaced  on  the  micrometer  parallel  to  itself,  the  linear  pattern 
moves  through  the  wide-slit  image  from  right  to  left;  or  the  reverse.  It  does 
so  also  when  either  mirror,  M  or  N,  is  slightly  rotated  on  a  vertical  axis.  The 
change  in  appearance  during  this  transfer  is  very  striking.  In  the  middle, 
between  the  extreme  right  and  left  positions,  the  linear  phenomenon  is  excep- 
tionally strong  and  fairly  tumbling  in  its  mobility.  Toward  the  right  or  left 
from  the  center  it  becomes  gradually  less  intense,  and  on  one  side  merges  into 
the  homogeneous  striations  which  then  appear.  On  the  other  side  it  seems 
merely  to  vanish.  Doubtless  the  linear  phenomenon  is  found,  as  usual,  at 
the  line  of  symmetry  of  two  reversed  spectra;  but,  as  both  spectra  are 
shrunk  to  very  small  lateral  dimensions,  many  colors  probably  adequately 
coincide.  In  an  achromatic  reproduction  of  the  slit  all  colors  will  coincide. 

It  is  thus  not  necessary  that  the  edges  of  the  slit  images  should  be  super- 
posed to  produce  the  linear  phenomenon.  What  is  still  more  curious  is  the 


86  THE   INTERFEROMETRY  OF 

result  that  not  even  the  longitudinal  axes  of  the  spectra  need  be  quite  in  coin- 
cidence, though,  of  course,  the  phenomenon  appears  most  intensely  for  the 
case  of  precise  superposition.  The  angle  of  admissible  separation  of  longitu- 
dinal axes  is,  however,  much  larger  here  than  in  the  usual  cases  above,  so  that 
one  of  the  longitudinal  guide-lines  of  the  two  spectra  may  be  appreciably 
above  the  other. 

The  last  result  and  the  fact  that  the  linear  phenomenon  appears  here  with 
an  indefinitely  wide  slit  are  new  features.  The  cause  of  the  latter  has  just 
been  referred  to  the  exceptionally  reduced  width  of  spectrum  resulting  from 
the  double  diffraction.  If  the  dispersion  were  quite  reduced  to  zero,  all  colors 
in  a  definite  narrow,  transverse  strip  of  the  white  slit  image  would  be  in  a  condi- 
tion to  interfere.  This  strip  contains  the  superposed  images  of  an  indefinitely 
fine  slit.  The  slit  in  any  other  position,  right  or  left,  would  have  two  non- 
coincident  images.  Hence,  when  one  wide-slit  image  moves  over  the  other, 
there  is  also  a  shift  of  the  linear  phenomena. 

To  produce  the  linear  phenomenon  with  sunlight  is  difficult.  The  inter- 
ferences should  first  be  produced  with  the  sodium  arc,  strongly,  and  the  arc 
thereafter  replaced  with  sunlight  entering  the  slit  at  the  same  angle.  Further- 
more, the  pencil  leaving  the  collimator  should  be  a  narrow,  vertical  blade  of 
light,  and  at  the  mirrors,  M  and  N,  red  and  green  light  should  be  screened 
off,  retaining  only  a  narrow  strip  of  yellow  light  for  each.  Finally,  to  avoid 
glare,  the  slit  is  not  to  be  too  broad  nor  too  narrow  to  cut  off  the  yellow  field 
of  the  telescope. 

Under  these  circumstances  of  completed  adjustment,  the  linear  phenomenon 
usually  appears  strongly.  Its  form  may  be  greatly  modified  by  rotation  of 
either  mirror,  M  or  N,  micrometrically,  around  the  vertical  axis,  as  already 
suggested.  The  types  are  given  in  figure  62,  quite  fine,  nearly  vertical  lines, 
q,  changing  to  moving,  coarser  forms,  m,  and  these  into  the  tumbling  variety, 
t,  very  coarse  and  nearly  horizontal.  The  latter  change  by  rotation  and  dimi- 
nution into  m'  and  q',  while  N  is  being  continually  rotated  over  a  very  small 
angle,  sliding  one  slit  image  continuously  over  the  other.  In  the  condition  t, 
the  fringes  rotate  with  astonishing  rapidity,  and  this  rotation  is  nearly  180°; 
i.e.,  if  the  angle  between  m  and  m'  is  a,  the  angle  of  rotation  has  been  180°-  a, 
so  that  between  q  and  q'  there  is  about  180°  of  rotation.  At  the  stage  /,  with 
fine  micrometric  adjustment,  the  fringes  may  be  made  quite  horizontal,  and 
they  are  then  relatively  large  and  square,  or  at  times  shaped  like  blunt  arrow- 
heads. This  rapid  rotation  of  fringes  near  t  accounts  for  their  turbulence, 
since  tremors  have  the  effect  not  merely  of  raising  and  lowering  them,  but 
also  of  producing  the  rotary  motion  in  question.  They  may  also  be  rotated, 
of  course,  on  slightly  tilting  the  grating  about  an  axis  normal  to  its  face. 
Rotating  the  latter  on  an  axis  parallel  to  its  face  places  the  phenomenon  in 
different  parts  of  the  superposed  yellow  field. 

Since  a  preponderance  of  yellow  homogeneous  light  is  present  in  the  whole 
of  the  superposed  wide-slit  images  in  the  telescope,  it  is  not  difficult  to  suggest 
the  cause  for  the  variations  of  the  interference  pattern  when  one  image  passes 


REVERSED   AND   NON-REVERSED   SPECTRA.  87 

horizontally  over  the  other.  The  forms,  t,  correspond  to  minimum  path- 
difference,  remembering  that  in  accordance  with  figure  59  all  rays  pass  the 
plate  of  the  grating  twice. 

Further  experiments  were  made  with  sunlight  to  detect  the  changes  which 
befall  the  phenomena  in  different  focal  planes.  The  ocular  of  the  telescope 
was  gradually  drawn  out  from  an  inner  extreme  position  to  an  outer  extreme 
position,  through  the  normal  position  for  principal  focal  plane.  In  this  case 
a  variation  of  form  corresponding  closely  to  figure  62  was  also  observed.  The 
characteristic  feature,  however,  was  the  prevalence  of  arrow-head  or  caret- 
shaped  lines,  both  in  the  case  of  the  extremely  fine  striations  and  of  the  coarser 
nodules.  In  the  former  case  these  roof -like  designs  were  closely  packed  from 
end  to  end  of  the  phenomenon  and  usually  pointed  upward.  They  recall  the 
top  edges  of  extremely  eccentric  ellipses  in  displacement  interferometry,  and 
in  view  of  their  lateral  motion  with  the  micrometer  M  and  the  decreased  dis- 
persion due  to  double  diffraction,  their  origin  may  be  similar. 

39.  The  same.  Inferences. — When  the  pencils,  Mm  and  Nn,  figure  59,  are 
parallel  and  sodium  light  is  used,  the  whole  field  is  uniformly  striated,  whether 
the  striations  are  made  fine  or  coarse.  I  have  found  it  impossible,  on  placing 
plate  compensators  (0.5  to  1.5  cm.)  in  both  beams  and  rotating  these  to  any 
degree  whatever,  to  produce  any  suggestion  of  a  secondary  periodicity  in  the 
field.  The  fringes  for  a  thick  compensator,  slightly  wedge-shaped,  merely 
become  a  little  finer.  Films  of  mica  are  liable  to  blur  the  field.  In  general, 
moreover,  reflections  would  be  relatively  weak  and  thus  inappreciable.  They 
would  require  a  separate  adjustment  for  coincidence  and  not  appear  with 
the  principal  phenomenon.  Hence  the  strands  of  interferences  obtained  in 
case  of  crossed  rays  are  in  a  measure  unique.  The  second  periodicity  is  not 
stationary,  but  a  part  of  the  phenomenon.  The  glass  plate  of  the  grating 
produces  an  effect  in  virtue  of  its  thickness,  precisely  as  in  the  case  of  the  dis- 
placement interferometry  of  my  earlier  papers. 

Experiments  made  with  polarized  light  proved  to  be  entirely  negative. 
The  phenomenon  appears  between  a  polarizer  and  an  analyzer  so  long  as 
sufficient  light  is  present  to  ex-  M 

hibit   it.      Observation    with    a  ty     71 

nicol,  in  the  absence  of  the  polar- 
izer, showed  nothing  but  the  ob-     -/ 
vious  effect  of  reflection.  & 


a 


63 


The  occurrence  of  these  par- 
allel strands  for  crossed  rays  and 
homogeneous  light  is  thus  diffi- 
cult to  explain.  I  have  tried  a  great  variety  of  methods  of  superposing 
special  interferences,  etc.,  to  produce  the  nodules  with  parallel  rays,  mM 
and  nN,  or  to  break  them  with  crossed  rays,  mN  and  nM,  without  avail. 
There  is  no  focal  plane  effect,  nor  any  polarization  effect.  It  is  therefore 
necessary  to  confront  the  case  at  its  face  value,  as  in  figure  63.  Here  S 


gg  THE   INTERFEROMETRY   OF 

and  5'  are  the  traces  of  two  longitudinally  coincident  reversed  spectra,  drawn 
apart  for  distinction,  the  region  of  the  D  lines  only  being  used.  The  light  is 
homogeneous  to  this  extent  and  the  slit  wide,  so  that  there  is  oblique  inci- 
dence. Then  every  point  of  5  should  (on  adjustment)  interfere  with  every 
point  of  5',  the  result  showing  a  uniformly  striated  field  in  the  telescope. 
This  is  emphatically  the  case  for  the  parallel  rays,  mM,  nN;  but  with  the 
crossed  rays,  mN,  nM,  the  interference  is  confined  to  the  rays  in  the  equi- 
distant positions,  n,  in  figure  63,  and  midway  between  them  the  field  is  a 
neutral  yellow.  In  other  words,  between  the  rays  n  the  rays  are  displaced, 
as  shown  by  the  arrows,  recalling  the  arrangement  of  nodes  in  acoustics. 

Corresponding  rays  a  and  a'  (for  instance)  do  not  coincide  and  hence  can 
not  interfere,  the  region  aar  remaining  neutral.  In  figure  64  the  rays  crossing 
at  c  (fig.  57)  have  been  shown  for  three  nodes  and  the  transverse  arrows  indi- 
cate the  directions  in  which  the  rays  have  been  urged  laterally.  Naturally,  I 
am  merely  stating  the  case  as  immediately  suggested  by  the  results.  One  may 
argue  that  there  may  be  a  secondary  periodicity  in  the  grating.  But  why 


65 


does  it  not  appear  at  all  in  the  case  of  parallel  pencils,  when  it  is  so  obtrusive  in 
the  case  of  crossed  pencils  of  rays?  Again,  the  interferences  are  unquestion- 
ably due  to  Di  and  L>2  light,  simultaneously.  If  the  grids  for  these  two  wave- 
lengths should  be  at  a  slightly  different  angle  to  each  other,  their  superposition 
would  give  something  like  the  observed  phenomenon,  apart  from  details.  Thus 
in  figure  65  the  two  grids  due  to  D\  and  D2,  intersecting  at  a  small  angle,  may 
be  interpreted  as  appearing  strand  or  cord  like  at  N,  and  neutral  at  /  and  I'. 
With  white  light  the  linear  phenomenon  would  eventually  become  achromatic. 
But,  again,  why  should  lines  so  close  together  as  DI  and  D2  show  any  appre- 
ciable difference  of  angle  or  rotational  phase-difference  in  their  interference 
pattern?  Intersecting  grids,  moreover,  can  be  produced  by  other  methods 
and  nearly  always  betray  their  origin.  The  final  inference  is  that  suggested 
by  figures  63  and  64,  that  homogeneous  rays  on  crossing  (here  in  a  medium 
of  plate  glass)  may  exert  a  lateral  influence  on  each  other,  to  the  effect  that 
identical  rays  emerging  from  the  crossing  are  arranged  in  equidistant  nodal 
planes  according  to  figure  63. 

40.  Experiments.  Reflecting  grating.  Crossed  rays.— In  the  preceding 
experiments  the  remarkable  phenomenon  of  double  interferences  was  ob- 
tained with  glass-plate  apparatus.  It  is  improbable  that  any  secondary  inter- 
ference can  have  been  produced  by  the  presence  of  reflected  light,  since  the 
reflected  pencils  will  be  weak  as  compared  with  the  primary  pencils  and 


REVERSED   AND   NON-REVERSED    SPECTRA.  89 

differently  situated.  It  is  nevertheless  necessary  to  forestall  all  misgivings 
by  avoiding  glass  plates  altogether  and  adapting  the  methods  of  figure  57, 
where  reflecting  surfaces  (front  faces)  only  are  present,  to  the  experiment 
for  crossed  rays  mN  and  nM. 

In  the  apparatus  as  finally  perfected,  G,  figure  57,  was  a  Michelson  plate 
grating  and  G'  a  Rowland  concave  grating,  each  with  about  the  same  grat- 
ing constant.  A  strong  lens  was  placed  at  T  for  observation  at  the  focus  of 
the  concave  mirror  of  G'.  The  latter  was  capable  of  fore-and-aft  motion,  of 
rotation  about  a  vertical  axis  in  its  own  plane  and  about  an  axis  normal  to  that 
plane ;  G  was  capable  of  rotation  about  a  horizontal  axis  parallel  to  its  plane. 
Thus  the  possibility  of  fore-and-aft  motion  and  the  three  cardinal  rotations 
for  the  gratings,  together  with  a  micrometric  fore-and-aft  motion  of  M ,  was  at 
hand,  as  well  as  the  rotation  of  M  and  N  about  horizontal  and  vertical  axes. 

The  interferences  were  found  after  establishing  the  coincidence  of  the  yellow 
homogeneous  fields,  in  the  manner  described  in  the  preceding  paragraph. 
The  fringes  were  at  first  small  and  apparently  single,  but  they  could  be 
enlarged  at  pleasure  and  the  two  definite  systems  separated  by  fore-and-aft 
motion  of  G'.  They  occupied  only  a  part  of  the  wide  yellow  slit  image,  the 
sodium  arc  being  used.  On  actuating  the  micrometer  at  M  there  was  dis- 
placement of  the  interference  pattern  as  a  whole,  so  that  the  conditions  of 
displacement  interferometry  are  here  also  implied,  though  the  equations  are 
liable  to  be  different.  On  rotating  M,  micrometrically,  about  a  vertical  axis, 
the  structure  of  the  interference  reticulations  changed  and  was  at  times 
reduced  to  a  single  set. 

Whenever  the  arc  flashed,  or  when  white  light  was  used,  the  linear  phenom- 
enon appeared  alone,  either  cross-hatched  or  longitudinal,  depending  upon 
the  character  of  the  reticulated  pattern  for  homogeneous  light.  With  sun- 
light, even  after  narrowing  the  blade  from  the  collimator  and  screening  off 
red  and  green  light,  the  phenomenon  was  faint  and  hard  to  find,  unless  it 
was  produced  alternately  with  sodium  arc. 

With  the  arc  freshly  charged  with  sodium,  but  a  single  set  of  interferences 
or  else  the  linear  phenomenon  appears,  since  the  broadened  sodium  lines  are 
equivalent  to  a  continuous  spectrum  in  this  region.  Not  until  the  excess  of 
sodium  has  all  been  evaporated  and  the  sodium  lines  are  normal  does  the  true 
reticulation  show  itself.  It  is  interesting  to  describe  two  cases  of  this  double- 
interference  pattern,  obtained  by  gradual  and  successive  fore-and-aft  motion 
of  the  grating  G',  between  limits,  while  the  edges  of  the  two  wide-slit  images, 
respectively  horizontal  and  vertical,  are  kept  in  contact  throughout. 

Suppose  the  original  fine  fringes  to  be  nearly  vertical;  then  the  apparently 
simple  fringes,  a,  figure  66  (their  appearance,  however,  would  lead  one  to  sus- 
pect their  simplicity),  change  to  the  cord-like  strands  b,  appearing  like  helices 
of  a  very  large  pitch.  Both  interference  fringes  are  still  nearly  parallel,  and 
they  cover  the  whole  wide-slit  image  uniformly.  These  eventually  pass  into 
the  square  or  rectangular  reticulation,  c,  with  both  systems  equally  strong. 
Probably  intermediate  forms  have  here  been  skipped.  The  system,  e,  occurs 


90 


THE   INTERFEROMETRY  OF 


very  soon  afterward,  in  which  the  difference  in  size  of  fringes  has  become 
enormous.  Following  e,  the  procession  is  reversed  in  g,  h. 

Both  systems  (a  and  0  systems,  say)  have  passed  through  maxima,  but 
not  at  the  same  time,  or  not  for  the  same  fore-and-aft  adjustment.  Both  sys- 
tems have  rotated,  the  rotation  being  very  rapid  near  the  maximum.  The 
reticulations  quiver  and  look 
precisely  like  capillary  waves 
in  a  rectangular  trough  of  mer- 
cury, except  that  they  are  usu- 
ally at  an  angle  to  the  bound- 
ing edges  of  the  superposed 
slit  images. 

In  this  quivering  system  of 
two  identically  strong  fringes 
it  is  difficult  to  make  out  the 
rotations,  but  after  consider- 
able revision  the  sequence  in 
figure  67  was  definitely  ascer- 
tained. Beginning  with  the 

extreme  fore-and-aft  position  of  G',  and  moving  it  successively  forward  in  steps 
of  i  or  2  millimeters,  the  apparently  single  grid,  i  changes  to  2,  where  the 
two  systems  a  and  /3  can  be  disentangled,  a  expanding  and  rotating  more 
rapidly,  so  that  3  and  4  follow.  Here  a  is  horizontal  and  probably  of  maxi- 
mum size,  /3  is  still  nearly  vertical  and  but  slightly  expanded.  Therefore,  while 
the  a-effect  wanes  the  /3-effect  waxes,  and  the  squared  or  orthogonal  type,  5,  is 
produced.  The  lines  are  here  equally  strong  and  it  is  the  symmetrical  figure 
of  the  series.  Thereafter  in  5,  6,  7,  8,  and  9  the  chief  expansion  and  rotation 
is  transferred  to  the  /3  system,  with  which  the  a  system  has  changed  functions. 
Hence  both  systems  rotate  nearly  180°  in  the  same  direction  and  pass  through 
maximum  size;  but  the  maximum  is  retarded  in  rotational  phase  for  one  as 
compared  with  the  other.  Rotation  and  growth  are  accelerated  near  the 
maximum.  The  total  displacement  of  the  grating  G'  between  the  cases  i  and 
g  (fig.  67)  was  about  2  cm. ;  but  this  depends  upon  the  obliquity  of  the  grating 
and  incidental  conditions,  as  explained  above. 

Suppose,  in  the  second  place,  that  the  original  fringes,  i,  figure  67,  were 
nearly  horizontal;  in  such  a  case  the  evolution  is  much  the  same,  but  the  sym- 
metrical form  number  5  becomes  smaller  and  more  and  more  flatly  rhomboidal 
horizontally.  Probably  the  scheme  of  rotation  is  the  same,  but  is  much  harder 
to  ascertain  in  view  of  the  flat  forms.  On  the  other  hand,  the  field  now  abounds 
in  vertical  strands  of  interferences,  like  those  of  the  preceding  paragraph,  and 
nodules  are  often  in  evidence,  as  before. 

If  the  original  lines  are  quite  vertical,  they  do  not  seem  to  rotate  with  fore- 
and-aft  motion  of  G',  but  form  intersecting,  vertical,  apparently  simple  sys- 
tems throughout  the  motion.  Slight  departure  from  the  vertical  produces 
rhomboids  very  long  vertically  and  often  very  coarse. 


REVERSED   AND   NON-REVERSED   SPECTRA.  91 

41.  The  same.     Compensators. — A  compensator  of  ordinary  plate  glass, 
at  the  intersection  c,  figure  57,  produces  no  effect,  if  symmetrical  to  both  beams. 
If  not  symmetrical,  the  interferences  are  displaced  to  right  or  left  in  the  field 
of  the  telescope,  as  in  any  case  of  displacement  interferometry,  depending  on 
which  component  beam  receives  the  longer  glass-path.    Thus  this  adjustment 
corresponds  to  the  grating  in  the  preceding  paragraph,  the  difference  being 
that  in  the  latter  case  the  same  ruling  is  used  for  both  diffractions.    Hence 
the  interference  figures  obtained  are  simpler,  showing  vertical  strands  only. 
In  the  present  case  strands  occur  in  all  directions.    The  maxima  for  oblique 
positions  of  the  glass  plate  were  not  found  with  reflecting  gratings. 

If  the  compensator  is  within  i  inch  in  thickness,  its  introduction  occasions 
no  difficulty.  The  interference  pattern  may  be  changed,  but  it  remains  the 
same  during  the  rotation  of  the  compensator;  but  if  the  latter  is  thicker  than 
2  inches,  the  figure  is  usually  so  small  as  to  be  found  with  difficulty,  unless 
the  grating  G'  is  brought  forward,  to  allow  for  the  mutually  inward  refraction 
of  the  rays.  If  this  is  done,  the  same  figure  may  be  reproduced.  On  advancing 
the  grating,  plate  compensators  much  over  3  inches  thick  were  tested  without 
the  slightest  annoyance.  Lenticular  compensators  require  special  adjust- 
ment and  are  very  difficult  of  use. 

The  effects  of  rotating  the  grating  about  the  three  cardinal  axes  have  all 
been  considered  above.  In  the  present  instance  two  sets  of  fringes  are  sym- 
metrically rotated,  subject  to  the  same  conditions.  Rotation  of  G'  around  a 
horizontal  axis  requires  an  elevation  or  depression  of  the  arc  lamp,  if  the 
fringes  are  to  remain  in  the  field.  Rotation  around  a  vertical  axis  separates 
the  slit  images,  and  a  readjustment  for  superposition  is  necessary.  Results 
so  obtained  are  therefore  complicated  and  were  not  studied. 

42.  Miscellaneous  experiments.     Fringes  with  mercury  light. — A    few 

random  experiments  made  with  the  sodium  arc,  in  the  presence  and  absence 
of  the  magnetic  field,  showed  no  results;  nor  was  this  to  be  expected,  as  a 
reasonably  strong  field  would  blow  out  the  arc.  Again,  the  insertion  of  a 
glass  compensator,  0.7  cm.  thick,  in  one  of  the  component  beams,  developed  no 
maximum  on  rotating  the  compensator  about  a  vertical  axis.  Thus  with  reflect- 
ing gratings  the  peculiar  behavior  of  the  transmitting  grating,  showing  a  maxi- 
mum on  either  side  of  a  symmetrical  minimum  (§36,  37),  is  not  reproduced. 
The  effect  of  rotating  the  first  reflecting  grating  G  on  a  vertical  axis  is  only 
to  throw  the  sodium  light  out  of  one  side  or  the  other  of  the  (superposed) 
slit  images.  No  available  means  of  enlarging  the  fringes  indefinitely  was 
found.  It  is  probable  that  this  would  require  fine  adjustment  for  symmetry. 
The  field  of  interference,  as  a  whole,  is  within  a  spot-like  area  which  may  be 
moved  up  and  down,  or  right  and  left,  by  the  vertical  and  horizontal  adjust- 
ment screws  on  the  mirror  M.  Coincidence  at  the  two  sides  of  the  slit  favor 
different  interferences.  The  case  is  always  as  if ,  at  a  single  point  of  the  field 
only,  there  were  actual  coincidence,  and  that  the  interference  pattern  is 
grouped  closely  around  it. 


92  THE   INTERFEROMETRY   OF 

With  the  use  of  an  ordinary  glass  mercury  lamp  (27  storage  cells,  5  amperes) 
the  fringes  are  found  with  difficulty  when  the  beam  at  the  first  grating  is 
wide.  On  using  a  vertical  blade  of  light  the  definition  was  improved.  The 
fringes  are  faint,  very  susceptible  to  motion,  and  at  times  even  absent.  They 
occur,  however,  as  a  single  set,  as  was  anticipated,  showing  that  the  above 
duplicated  fringes  are  actually  due  to  the  two  sodium  lines.  The  mercury 
fringes  are  easily  rotated  and  pass  through  a  horizontal  maximum  with  fore- 
and-aft  motion.  Rotating  G  about  a  normal  axis  may  further  increase  this 
maximum  size  to  a  limit  at  which  the  fringes  appear  irregular  or  sinuous.  A 
displacement  of  the  mirror  M  over  0.7  cm.  was  easily  permissible,  without 
destroying  the  fringes.  They  occur,  as  above  stated,  within  a  certain  adjusted 
spot  area  of  the  field  of  view.  An  attempt  was  again  made  to  detect  a  Zeeman 
effect  by  placing  the  poles  of  an  electromagnet  on  the  two  sides  of  the  lamp; 
but  here  again  no  difference  was  discernible  on  opening  and  closing  the  electric 
circuit.  The  field,  however,  for  incidental  reasons,  could  not  be  made  strong 
enough  for  a  critical  experiment. 

43.  Inferences. — After  these  experiments  (made  with  the  apparatus  figure 
57,  free  from  glass  plates  and  depending  on  reflections  only)  the  cause  of  the 
phenomenon  is  no  longer  obscure.  Obviously  one  of  the  paired  grids  in  figure 
66  or  67  belongs  to  each  sodium  line.  The  retardation  of  one  phenomenon, 
rotationally,  as  compared  with  the  other,  is  due  to  the  difference  in  wave- 
length between  Di  and  Dz.  The  phase-difference  between  numbers  4  and  6 
(fig.  67)  is  thus  equivalent  to  6  Angstrom  units.  If  the  displacement  of  G' 
is  about  0.3  cm.,  there  should  be  about  0.5  mm.  displacement,  fore  and  aft, 
for  i  Angstrom  unit.  If  the  grating,  G',  is  on  a  micrometer,  this  should  be  a 
fairly  sensitive  method  of  detecting  small  differences  of  wave-length,  or  give 
evidence  of  doublets  lying  close  together.  The  sensitiveness  clearly  increases 
with  the  length  of  path  of  the  component  rays  and  may  thus  be  increased. 

With  this  definite  understanding  of  the  phenomenon,  it  is  desirable  to  deduce 
the  equations,  which  in  the  occurrence  of  parallel  rays  would  not  differ  essen- 
tially from  those  of  Chapter  II  or  III.  It  is  useful,  however,  to  treat  the  new 
case  of  crossed  rays.  In  figure  69  the  angles  of  diffraction  are  0i  and  02,  if  the 
incidence  of  light,  L,  is  normal  at  G  and  at  an  angle  iz  at  G't  G  and  G'  being 
parallel.  The  mirrors  are  set  symmetrically  at  angles  ffi  and  0-2  to  the  normal 
in  question,  and  the  diffracted  rays  are  reflected  at  angles  0:1/2  and  «2/2, 
respectively.  The  reflected  rays  cross  the  normal  at  an  angle  £.  Then 

sin  0i  =  \/D i  sin  02  =  \/D2  -  sin  it 

where  DI  and  D2  are  the  grating  constants.    From  the  figure 

«l/2  =  0i+<r  1-90°  a2/2  =  02  +  0-2-90°  81=  di  + 0  02  =  «2  +  /3 

From  these  equations, 

Disini=D2  sin  (202+2(T2+/3)-D1  sin  (-...) 
If  Di  =  D2,  then  0i=02,  al=ffz,  and  therefore  t2  =  o. 


REVERSED   AND   NON-REVERSED    SPECTRA. 


93 


Thus  the  relations  are  quite  complicated,  but  if  Di  =  Dz,  or  the  gratings 
have  the  same  constant,  rays  of  all  wave-lengths  should,  after  double  diffrac- 
tion, issue  normally  to  the  grating  Gf,  and  the  arrangement  is  therefore 
achromatic.  If  Di  is  not  quite  the  same  as  D2,  but  nearly  so,  an  adjustment 
of  a  would  probably  meet  the  case  approximately.  If  the  original  incidence  is 
at  an  angle  i\,  D\  sin  t'i  would  have  to  be  subtracted  from  the  first  member, 
but  the  diffractions  would  now  differ  on  the  two  sides  of  the  apparatus. 

The  relations  of  the  rotations  of  the  striations  of  DI  and  D2  light  to 
the  fore-and-aft  motion  is  next  to  be  considered.  It  will  be  convenient  to 
make  use  of  figure  68  for  this  pur- 
pose, the  notation  being  the  same 
as  in  figure  69.  The  two  rays,  i 
and  2  (Di  and  Dz),  have  both 
been  introduced,  and  the  position 
of  G'  is  "such  that  the  D2  rays 
intersect  in  its  face  and  are  dif- 
fracted into  Tz.  In  such  a  case 
the  combined  pencil  is  divergent, 
DI  rays  will  undergo  an  earlier  £> 
intersection,  and  consequently  be 
separately  diffracted  into  T\  and 
T\.  Hence  DI  and  D2  are  differently  circumstanced  in  relation  to  the  fore- 
and-aft  motion,  and  the  rotation  produced  will  thus  be  advanced  in  one 
case,  as  compared  with  the  other,  for  the  reason  discussed  in  Chapter  III, 
paragraph  26.  It  is  also  clear  that  the  difference  of  phase  in  the  two  rotations 
will  be  greater,  as  the  total  path  of  rays  between  G  and  G'  is  greater,  so  that 
the  large  distances  used  in  the  present  experiments  (nearly  3  meters)  account 
for  the  astonishing  sensitiveness  of  the  phase  of  rotation  to  the  wave-length 
difference.  In  fact,  D2  will  be  in  the  same  phase  as  DI,  if  the  grating  is  moved 
forward  from  G'  to  g',  figure  68,  since  in  both  cases  the  rays  intersect  in  the 
normal.  Hence  if  R  is  the  total  path  GmNG\  and  if  the  angle  of  dispersion 
between  DI  and  D2  is  dd,  02  the  angle  of  diffraction  at  G',  and  h  the  displace- 
ment from  G'  to  g',  D  the  grating  space, 


69 


=  hsin  02 


and  the  resolving  power 


,fl    fcx 
*'*DR 


d\ 


Dh 


X     R  cos  02    RI/  Z)2_x2 
In  the  given  adjustment,  roughly, 

dQ  =  3.'jXio~4  R  =  3oocm.  02  =  2oc 


whence 


- 


_ 

0.34 


94  REVERSED  AND   NON-REVERSED    SPECTRA. 

As  the  resolving  power  is,  roughly,  h/R,  and  if  h  =  0.003  cm.  is  still  appreciable, 


3Xio2 

i.e.,  lines  i/ioo  of  the  distance  apart  of  the  sodium  lines  should  be  rotationally 
separated. 

Again,  the  displacement,  fore  and  aft,  between  like  rotational  phases  of 
Di  and  Dz  should  be  about  3  mm.,  and  this  agrees  fairly  well  with  the  order 
of  values  found. 

The  case  of  the  transmitting  grating  (fig.  59)  is  thus  also  elucidated,  though 
it  is  not  clear  to  me  why  the  duplication  of  fringes  is  so  efficiently  concealed 
in  the  nodular  forms  observed.  The  reason  for  the  minimum  of  size,  for  the 
symmetrical  position  i  =  o,  and  the  two  maxima  for  oblique  positions  of  the 
grating  (^=±20°  about),  suggests  an  explanation  similar  to  that  given  in. 
Chapter  II.  In  other  words,  in  the  oblique  position  the  short  path-length 
is  compensated  by  the  increased  thickness  resulting  from  the  greater  obliquity 
of  grating,  whereas  the  long  path-rays  traverse  the  plate  of  the  grating  more 
nearly  normally.  In  this  way  the  path-difference  is  reduced  as  compared 
with  the  symmetrical  position,  and  the  fringes  are  therefore  larger.  The 
oblique  grating  acts  as  a  compensator  in  both  of  the  component  beams,  and  the 
fringes  may  be  visible,  even  if  in  the  original  position  (fig.  59)  they  are  all  but 
invisible.  If,  however,  the  apparatus  (fig.  57)  is  used  with  a  plate-glass  com- 
pensator symmetrical  at  c,  there  are  no  maxima  or  minima  for  any  obliquity. 
Hence  the  tentative  explanation  for  the  case  of  figure  59  is  not  warranted. 

The  fore-and-aft  motion  of  the  plate  grating  (fig.  59)  produces  no  effect, 
since  the  rays  are  reflected  back  so  as  to  retrace  their  paths.  They  are  also 
reflected  between  parallel  mirrors  N,  m  and  n,  M.  Thus  the  path-difference 
is  not  modified.  The  result  is  merely  a  decrease  of  the  distance  M,  N,  and  a 
corresponding  increase  of  m,  n,  and  vice  versa. 

The  marked  effects  produced  by  rotating  the  transmitting  grating  around  a 
normal  axis,  finally,  follow  the  explanations  given  for  the  rotation  of  fringes 
of  non-reversed  spectra  in  Chapter  III,  paragraphs  25  and  26. 

In  conclusion,  an  interesting  application  of  the  apparatus  (fig.  56)  or  the 
other  similar  types  may  be  suggested.  By  half-silvering  the  mirrors  and  pro- 
viding a  similar  opaque  set  beyond  them,  there  should  be  no  difficulty  (in 
the  case  of  homogeneous  light)  of  bringing  the  interferences  due  to  crossed  rays, 
c,  and  to  parallel  rays,  a'b',  into  the  field  of  the  telescope  together.  Strictly 
homogeneous  light  (mercury  arc)  would  be  needed  to  obviate  the  duplication 
of  the  sodium  arc.  In  such  a  case,  therefore,  the  parallel  fringes  could  be  used 
after  the  manner  of  a  vernier  on  the  crossed  fringes,  with  a  view  to  a  repetition 
of  the  experiment  of  Michelson  and  Morley,  if  this  experiment  had  not  been 
so  thoroughly  carried  out  by  the  original  investigators.  However,  the  plan 
would  be  to  rotate  the  apparatus,  as  a  whole,  so  that  the  two  crossed  rays 
would  be  alternately  in  and  at  right  angles  to  the  earth's  motion,  whereas  the 
two  parallel  rays  would  preserve  the  same  relation  to  that  motion.  Naturally, 
the  parallel  and  crossed  paths  would  in  such  a  case  have  to  be  lengthened  by 
multiple  reflections. 


CHAPTER  VI. 


CHANNELED    SPECTRA  OCCURRING  IN  CONNECTION  WITH    THE  DIFFRAC- 
TIONS OF  REFLECTING  GRATINGS. 

44.  Introductory. — Throughout  the  preceding  work  I  had  noticed  that 
the  spectrum  due  to  either  of  the  component  beams,  after  successive  reflection 
from  two  reflecting  gratings,  was  often  regularly  furrowed  by  transverse 
black  bands,  before  the  two  spectra  were  brought  to  interfere.     As  these 
fringes  are  stationary,  they  do  not  modify  the  phenomenon  investigated;  but 
questions  now  arise  as  to  whence  these  reflected  fringes  of  a  single  beam  come. 
They  are  not  strong,  as  a  rule,  and  I  was  therefore  inclined  to  attribute  them 
to  some  imperfection  of  the  silvering  of  the  opaque  mirrors,  but  this  proved 
not  to  be  the  case,  so  that  it  seemed  worth  while  to  examine  them  by  special 
experiments. 

45.  Apparatus. — The  apparatus  for  this  purpose,  as  one  beam  only  is 
wanted,  is  quite  simple.    In  figure  70,  L  is  a  vertical  blade  of  parallel  rays  of 
white  light  from  a  collimator  and  slit.  These  rays  impinge  on  the  plane  grat- 
ing G,  whence  the  orders  1,2, 

3 ,  etc. ,  of  spectra  are  reflected . 
Either  of  these  pencils  maybe 
received  by  the  second  grat- 
ing G',  plane  or  concave,  from 
which  spectra  of  any  order 
are  available.  If  o  denotes  the 
reflected  pencils,  the  groups 
from  two  gratings  may  be  dis- 
tinguished as  (3,  i),  (3,  o), 
(3.  -i),  (3.  -2),  etc.,  as  in  JQ 

the  figure.  Any  of  these  two 
different  pencils  is  to  be  examined  at  T  by  a  lens  or  telescope,  for  instance, 
and  the  latter  (with  strengthened  objective  where  needed)  is  more  convenient, 
even  when  the  concave  grating  is  used.  A  wide  slit  5,  revolvable  about  G, 
is  often  useful  for  screening  off  spectra  or  parts  of  spectra.  In  some  experi- 
ments the  grating  G'  may  be  replaced  by  an  opaque  mirror. 

The  gratings  are  provided  with  the  usual  adjustments  for  parallelism  of 
rulings  and  slit.  G'  and  T  must  be  capable  of  considerable  right-and-left 
motion,  and  G,  in  particular,  of  controllable  fore-and-aft  motion. 

46.  Scattering. — An  interesting  result  of  this  work  is  the  evidence  and 
spectroscopic  quality  of  scattered  rays,  incidentally  encountered.    For  instance 
in  figure  70,  if  the  slit  5  is  narrow,  it  cuts  off  all  the  rays  but  the  orange  yellow 
of  the  third  order,  and  the  reflected  spectra  (3,  i),  (3,  -i),  etc.,  will  largely 

95 


96  THE   INTERFEROMETRY   OF 

consist  of  orange-yellow  light.  Associated  with  each  of  these  reddish-yellow 
patches,  however,  are  vividly  violet-blue  patches,  each  separated  from  the 
reddish  yellow  by  an  almost  total  absence  of  green,  relatively  speaking.  If 
the  light  is  very  intense,  the  connecting  part  of  the  spectrum  also  appears, 
but  it  is  always  far  less  vivid  than  the  ends  of  the  spectrum  in  question. 

Inasmuch  as  all  violet  radiation  proper  has  been  screened  off  at  5,  it  is 
obvious  that  violet  light  must  have  been  scattered  in  all  directions  from  G, 
a  part  of  which,  therefore,  passes  the  slit  and  is  resolved  by  the  second  grating 
G'.  Moreover,  as  the  scattering  lines  of  the  grating  are  equidistant,  the 
scattered  light  has  a  regular  wave-front.  (Cf.  Cam.  Inst.  Wash.  Pub.  No. 
229,  1915,  pp.  100—102.) 

The  correlative  experiment  of  detecting  the  reddish  light  transmitted  after 
scattering  was  also  tested.  For  this  purpose  the  reflecting  grating  G  may  be 
replaced  by  a  transmitting  grating,  slit  5  placed  beyond,  and  the  light  then 
analyzed  by  a  second  grating  G'  behind  the  slit  and  diffracting  toward  it  on 
one  side.  But  no  results  of  value  were  obtained. 

47.  Fringes  with  white  light.— The  experiments  with  the  apparatus  (fig.  70) 
were  commenced  with  sunlight  and  (what  is  essential)  a  fine  slit.  Fringes 
are  found  in  all  combinations  of  doubly  diffracted  pencils  (3,  +i),  (3,  —  i), 
(3,  -2),  etc.;  (2,  i),  (2,  -i),  etc.;  (i,  -i),  (i,  i),  etc.,  but  none  in  the 
reflected  pencils  (3,  o),  (2,  o),  (i,  o),  etc.,  as  a  rule.  Whether  the  grating  G' 
be  concave  or  plane,  it  is  best  to  use  a  telescope  at  T,  because  (when  provided 
at  the  objective  with  an  auxiliary  concave  or  a  convex  lens)  it  more  easily 
offers  a  wide  range  of  observation  along  its  axis  than  an  ocular.  The  latter 
must  be  wide  and  has  to  be  shifted  bodily;  but  both  methods  were  used.  A 
concave  grating  at  G  and  plane  grating  at  G'  gave  no  results.  The  concave 
grating  is  usually  more  free  from  channeled  spectra. 

Of  the  great  variety  of  fringes  obtained,  I  shall  give  only  two  typical 
examples.  The  second  order  of  spectra  for  G  (plane)  was  separated  from  the 
others  by  the  slit  S  and  diffracted  into  G'  (fig.  70).  The  successive  fringes 
appear  as  the  ocular  is  drawn  outward  from  the  principal  focus. 

Combination  £2,  G'o :  Only  a  good  sodium  doublet,  which  became  washed 
on  drawing  out  the  ocular  of  T,  was  obtained;  no  fringes  appeared. 

Combination  £2,  G'—  i:  Just  outward  from  the  principal  focus  a  large, 
coarse,  irregular  set  of  fringes  appeared;  next  (ocular  farther  out)  a  large 
regular  set,  somewhat  diffuse,  possibly  double  and  superposed;  then  a  finer, 
half-size,  very  regular  set,  possibly  decreasing.  After  this  the  mottled  sur- 
faces of  the  gratings  were  successively  in  focus.  A  weak  spectacle  lens  was 
now  added  to  the  objective  of  T,  whereupon  very  large  regular  fringes  were 
seen  when  the  ocular  was  far  out. 

Combination  £2,  G- 2:  The  ocular  moving  outward  from  the  principal 
focus,  the  fringes  seen  in  succession  were  as  follows:  large,  regular,  vague; 
half-size  sharp;  surfaces  vertically  striated;  (lens  on)  fine  regular  set  in  red; 
doubled  regular  set  in  green. 


REVERSED   AND   NON-REVERSED    SPECTRA.  97 

Combination  £2,  G—  3 :  Fine  set  just  before  the  surfaces  appeared,  which 
were  delicately  striated;  fine  regular  set;  coarse  set,  both  close  to  surface; 
(with  lens  on)  fine  regular  set;  doubled,  strong  regular  set. 

Different  distances  between  G  and  G'  had  very  little  influence  on  the  size 
of  the  phenomena.  A  few  examples  may  be  given,  which  are  observed  when 
the  ocular  is  moved  outward. 

6*3,  Gi :  Distance  10  cm. — Fringes,  faint  regular;  strong  irregular;  faint 
regular;  flat  field;  surfaces  visible;  faint  regular. 

Distance  25  cm. — Strong  irregular;  faint  regular;  small  regular;  large 
(double)  irregular;  lines  slit  into  fine  fringes;  large  faint  regular. 

Distance  46  cm. — Large  strong,  with  two  absorption  bands;  fine  regular; 
double-sized  faint ;  surfaces  with  fine  striations ;  alternations  of  fine  and  coarse 
lines;  faint,  regular,  large,  etc. 

Fringes  of  different  color  are  often  in  different  focal  planes.  When  a  lens 
is  used  with  the  concave  grating,  observations  must  sometimes  be  made  2 
meters  off  to  get  the  large  regular  fringes.  Red  fringes  may  be  narrower  than 
the  corresponding  violet  set. 

If  the  grating  G  is  moved  fore  and  aft,  parallel  to  itself,  the  fringes  are 
shifted  across  the  stationary  sodium  line,  as  in  displacement  interferometry. 

Whereas  in  the  positive  combination  (3,  i),  (3,  2),  etc.,  the  spectra  widen, 
they  tend  to  close  up  for  the  negative  combinations  (3,  —  i),  (3,  —2),  etc. 
With  two  identical  plate  gratings  they  may  image  the  white  slit.  But  this 
seems  to  have  little  effect  on  the  fringes  seen  as  a  whole  when  the  ocular  is 
out  of  focus. 

When  white  light  is  used  and  the  grating  G'  replaced  by  an  opaque  mirror, 
or  in  case  of  combinations  which  involve  direct  reflection  (2,  o;  3,  o;  etc.)  at 
G',  there  seem  to  be  no  fringes. 

48.  Fringes  with  sodium  light. — While  there  is  some  difficulty  in  obtaining 
the  fringes  with  white  light,  fringes  with  homogeneous  light  are  obtained  at 
once,  provided  the  light  is  sufficiently  intense.  A  sodium  arc  lamp,  or  a 
mercury  lamp,  with  a  fine  slit,  must  therefore  be  used.  In  this  case,  moreover, 
the  grating  G'  may  often  be  replaced  by  an  opaque  mirror,  or  the  fringes  of 
the  order  £2  G'o,  GzGo,  etc.,  may  be  produced  with  entire  success.  On  moving 
G  fore  and  aft,  they  again  travel  across  the  sodium  line.  Often,  in  fact,  two 
sets  of  fringes  seem  to  be  shifted.  A  few  examples  again  may  be  given  of 
the  great  variety  in  this  display  while  the  ocular  is  being  drawn  out : 

Gi,G'  —  2~.  Sodium  lines  D\DZ  single  size;  large  strong  fringes,  lines  split. 

Gi,  G'—  i :  Closed  spectrum;  striations  continuous. 

Gi,  G'o:   Reflection;   D\Dz   single   size;   surfaces   of   gratings   finely 

striated. 

Gi,  G'i:  DJDz  double  size;  strong  grid  seen  very  near  the  surface  of  G'. 
Gi,  G'i :  DiDz  treble  size,  out  of  reach. 
G2,  G'o:   Reflection;  no  fringes. 


98  THE   INTERFEROMETRY   OF 

G2,G'—i:   Distance  12  cm. — Coarse  irregular;  with  lens  fine  regular 

set,  near  and  beyond  the  surfaces. 
£2,  G'—i:    Distance  45  cm. — Surfaces  with  doubled  fine  striations; 

with  lens  finally  strong  and  regular. 
£2,  G'—i:    Distance  60  cm. — Regular  faint;  irregular  double,  very 

strong;  surfaces  striated;  with  lens  strong  double  irregular; 

finally  regular  small. 
G$,  G'I:    Regular;  regular  line  split;  irregular  coarse;  surfaces  finely 

striated,  G  coarser;  fringes  grow  continually  larger  without 

vanishing. 

On  moving  G  fore  and  aft,  two  grids  seem  to  travel  through  each  other  in 
opposite  directions.  This  probably  accounts  for  the  occurrence  of  irregular 
fringes.  The  size  of  fringes  seems  to  be  a  minimum  for  a  conjugate  focus 
near  the  surfaces.  The  whole  phenomenon  is  continuous.  Irregular  fringes, 
probably  superpositions,  become  regular  in  other  focal  planes. 

£3,  G'z:  About  the  same;  minimum  size  at  the  surfaces,  increasing 
about  three  times  as  the  ocular  is  drawn  either  way. 

£3,  G'o;  also  £3,  mirror:  About  the  same  results,  only  brighter  and 
better.  Hence  in  case  of  large  dispersion  two  gratings  are  not 
needed.  The  two  sodium  lines,  when  the  ocular  is  drawn  out 
of  focus,  multiply  themselves  at  regular  intervals,  so  that  the 
grids  are  sometimes  distinct,  sometimes  partially  superposed. 
Thus  the  classic  diffraction  phenomena  of  a  slit  suggest  them- 
selves as  the  starting-point  for  an  explanation  of  the  present 
phenomena  as  a  whole. 

£3.  G'o,  produced  alternately  with  sodium  light  and  sunlight,  showed 
the  same  sequence  of  fringes  (the  large  ones  with  a  tendency 
to  split)  in  the  former  case,  while  nothing  appeared  in  the  case 
of  white  light. 

49.  Grating  on  a  spectrometer.— It  seemed  necessary,  therefore,  to  con- 
sider the  diffraction  of  a  fine  slit,  when  seen  in  the  telescope,  somewhat  in 
detail.  In  Chapter  III  the  production  of  beautiful  Fresnellian  interferences 
from  two  identical  slit  images  and  homogeneous  light  was  demonstrated;  but 
an  equally  clear  manifestation  of  the  diffraction  of  a  slit  image,  when  the 
ocular  is  out  of  focus,  does  not  seem  to  occur.  The  broad  image  of  the  slit 
out  of  focus  shows  a  stringy  structure  only,  but  no  separation  is  easily  obtain- 
able. Fringes,  as  such,  are  quite  absent  when  the  ocular  is  drawn  out. 

The  light  of  the  sodium  arc  was  now  passed  through  a  very  fine  slit  and 
collimator  and  reflected  from  a  plate  grating.  The  above  intermittently 
regular  and  irregular  fringes  were  strikingly  obtained  with  the  ocular  out  of 
focus.  As  this  is  successively  more  and  more  drawn  out,  fine  lines  become 
coarser,  and  then  seem  to  subdivide,  giving  the  structure  a  fluted  appear- 
ance, frequently  regular.  There  is,  in  other  words,  a  double  periodicity.  In 


REVERSED   AND   NON-REVERSED   SPECTRA.  99 


the  case  of  highly  diffracting  grating  (D==  lo^Xiys),  the  results  appear  best 
in  the  second  order. 

The  same  beautifully  duplicated  fringes  were  obtained  with  a  transmitting 
film  grating  of  about  the  same  dispersion,  particularly  well  in  the  first  order. 

The  sodium  flame  gives  too  little  light  for  the  present  purposes,  but  the 
phenomenon  is  seen. 

Believing  that  some  irregularity  might  be  introduced  by  the  double-sodium 
line,  I  installed  a  mercury  lamp  for  comparison.  In  the  first  experiment  a 
film  grating  (D=  173  X  io~*)  was  used,  the  ocular  traveling  outward  from  the 
principal  focus.  Both  the  green  and  the  double  yellow  mercury  lines  enlarged 
and  showed  fringes  of  increasing  size  and  number  together.  The  green  field 
had  a  darker  band,  the  yellow  a  bright  band  in  the  middle.  As  the  fringes 
enlarged,  each  split  up  into  secondary  fringes,  4  or  5  eventually,  and  this 
again  occurred  for  both  green  and  yellow  fields. 

Rotating  the  grating  around  a  vertical  axis  seemed  to  shift  the  primary 
fringes  laterally  over  the  stationary  secondary  fringes.  A  concave  lens  for 
positions  anterior  to  the  principal  focus  and  a  convex 
lens  for  posterior  positions  (toward  the  eye)  were 
successively  added  to  increase  the  range  of  observa- 
tion. On  both  sides  of  the  principal  focal  plane 
(fig.  71)  fringes  occur,  which  enlarge  with  the  dis- 
tance x  from  that  plane.  As  they  enlarge,  each  fringe 
splits  up  into  secondary  fringes,  which  in  turn  enlarge. 
Sometimes  the  arrangement  is  irregular.  Green  and  yellow  fields  may 
overlap,  but  they  do  not  do  so  conformably. 

The  undeviated  ray,  however  fine  the  slit  may  be,  merely  shows  a  stringy 
field,  sometimes  suggesting  structure,  but  never  showing  clear-cut  fringes. 

The  same  kind  of  results  were  obtained  with  a  reflecting  grating  of  about 
the  same  dispersive  power.  In  the  second  order  the  fringes  were  particularly 
clear  and  regular.  Primary  fringes,  finally,  carried  three  to  four  secondary 
fringes  each. 

Next,  a  ruled  transmitting  grating  of  less  dispersive  power  (grating  constant 
352  X  icr6  cm.)  was  adjusted  for  mercury  light.  Here  in  the  undeviated  ray 
and  in  the  first  order  no  clearly  separated  fringes  were  obtained.  In  the  second 
and  third  orders,  however,  they  were  very  perfect,  and  followed  the  above 
rules,  showing  sharp  secondary  fringes. 

It  follows,  therefore,  that  a  certain  degree  of  dispersion  is  needed  to  resolve 
the  fringes,  which  is  inadequate  in  amount  in  the  order  zero,  in  this  case, 
and  scarcely  so  in  the  first  order.  In  the  higher  orders  the  conditions  are  met. 
Using  a  very  fine  slit,  however,  I  later  just  succeeded  in  separating  the  fringes 
in  the  first  order. 

Finally,  I  returned  to  the  endeavor  of  detecting  diffraction  fringes  in  the 
undeviated  image,  using  a  micrometer  slit,  a  good  achromatic  lens  (or  no  lens), 
and  a  distant  (2  meters),  moderately  strong  telescope.  In  this  case  separated 
and  distinct  diffraction  fringes,  white  throughout,  were  undoubtedly  obtained. 
They  moved  with  the  eye  so  as  rarely  to  be  stationary  and  in  the  same  direc- 


100  THE   INTERFEROMETRY   OF 

tion  if  the  ocular  is  drawn  out,  or  the  reverse  if  it  is  thrust  in.  On  close 
examination  two  sets,  in  different  focal  planes,  seemed  to  be  present,  one 
stationary  and  the  other  moving  as  described,  and  accounting  for  the  observed 
pronounced  parallax.  Suggestions  of  movable  fringes  accompanying  the 
stationary  are  also  present  when  the  latter  are  produced  by  the  grating.  In 
this  case  the  stationary  fringes  are  strong;  in  the  case  of  simple  diffraction  the 
movable  fringes  are  more  prominent. 

50.  Inferences. —  There  can  be  no  doubt  that  the  great  variety  of  chan- 
neled spectra  obtained,  when  white  light  is  successively  diffracted  by  two 
gratings,  is  referable  to  the  fringes  obtained  in  the  diffraction  of  homogeneous 
light,  observed  outside  the  principal  focal  plane,  on  a  spectrometer.  In  other 
words,  if  light  of  a  given  pure  color  (sodium,  mercury)  is  used,  a  single  grating 
suffices.  Each  line  of  the  spectrum  is  resolved  into  well-defined  groups  of 
fringes,  if  it  is  observed  either  in  front  of  or  behind  the  principal  focal  plane. 
The  arrangement  of  fringes  varies  in  marked  degree  with  the  distance  of  the 
plane  observed  from  the  latter  (x,  fig.  71).  If  reflecting  gratings  are  used, 
there  is  no  other  possible  source  of  interferences ;  but  reflecting  and  transmit- 
ting gratings  show  the  phenomenon  equally  well. 

After  finding  how  easily  the  Fresnellian  interferences  of  two  virtual  slits 
could  be  reproduced  in  the  telescope  (Chapter  III)  and  observed  on  either 
side  of  (before  or  behind)  the  sharp  images,  it  seemed  reasonable  to  suppose 
that  the  diffraction  of  a  slit  could  also  be  produced  and  exhibited  in  this  way; 
but  the  availability  of  this  anticipation  is  attended  with  much  greater  diffi- 
culty. The  image  of  a  very  distant  slit  does  indeed  show  separated  diffraction 
fringes  on  either  side  of  the  principal  focal  plane  in  the  observing  telescope. 
But  they  move  right  and  left  with  the  eye,  in  the  same  direction  if  the  ocular 
is  drawn  outward  from  the  principal  focal  plane,  and  in  the  direction  opposite 
to  the  eye  if  the  ocular  is  thrust  in.  Hence,  in  this  respect,  the  fringes  do  not 
at  once  recall  the  phenomena  under  consideration.  Usually  the  blurred  image, 
out  of  focus,  is  stringy,  without  definite  structure.  It  is  resolved  in  a  single 
focal  plane  only. 

To  obtain  sharp  stationary  fringes  from  an  image  of  the  slit,  this  image  must 
be  produced  by  the  diffraction  of  a  grating  having  a  dispersing  power  above  a 
certain  minimum.  Thus  in  a  grating  of  about  7,000  lines  to  the  inch  the  un- 
deviated  slit  image  and  the  image  of  the  first  order  are  not  clearly  resolved, 
unless  the  slit  is  very  fine.  In  the  second  and  higher  orders,  however,  the  res- 
olution is  very  pronounced  and  the  fringes  stationary. 

The  resolution  of  fringes  is  equally  manifest  in  front  of  or  behind  the  prin- 
cipal focal  plane,  so  that  if  a  weak  convex  lens  is  added  to  the  objective  of  the 
telescope,  the  succession  of  fringes  is  found  with  an  outgoing  ocular;  if  a  weak 
concave  lens  is  added  to  the  objective,  the  succession  is  found  with  an  ingoing 
ocular,  starting  in  each  case  near  the  principal  focus.  As  the  fringes  increase 
in  size  they  in  turn  subdivide,  sometimes  irregularly,  as  if  each  fringe  were  a 
new  slit  image,  capable  of  undergoing  secondary  diffraction.  Beyond  these 
secondary  fringes  no  further  resolution  was  detected. 


REVERSED   AND   NON-REVERSED   SPECTRA.  101 

Returning  to  the  work  with  two  successive  gratings  and  white  light,  the 
channeled  spectra  obtained  are  too  complicated  for  concise  description.  A 
very  interesting  result,  however,  is  the  passage  of  the  fringes  across  the  sta- 
tionary sodium  line,  when  the  first  grating  G  is  moved  fore  and  aft  in  a  direc- 
tion normal  to  its  plane.  The  region  of  the  D  line  is  thus  alternately  dark  and 
bright.  The  direction  of  these  rays  remains  unaltered  while  the  illumined  strip 
is  shifted  horizontally  across  the  ruled  space  (fig.  70)  of  the  second  grating. 
Usually  it  is  difficult  to  see  the  D  line  in  the  focal  plane  of  the  fringes.  When 
homogeneous  light  is  used  this  fiducial  mark  is  necessarily  absent  and  the 
cross-hairs  of  the  ocular  must  be  supposed  to  replace  it.  The  shift  of  the 
fringes  is  then  equally  obvious,  and  sometimes  (sodium  light)  different  groups 
seem  to  travel  in  opposite  directions  while  the  grating  G  moves  in  one  direction. 
In  case  of  homogeneous  light  and  two  gratings,  moreover,  the  fringes  seem  to 
be  of  minimum  size  in  the  conjugate  focal  plane  of  the  gratings.  They 
increase  in  size  and  in  turn  split  up  in  focal  planes  before  and  behind  this. 

An  insight  into  these  occurrences  was  finally  obtained  in  observation  with 
homogeneous  light  in  the  spectrometer  by  shifting  the  grating  (transmitting) 
in  its  own  plane,  right  and  left.  The  fringes  in  such  a  case  move  bodily  across 
the  field  of  the  telescope,  new  groups  entering  on  one  side  for  those  which 
leave  on  the  other.  These  fringes,  even  if  quite  distinct,  are  differently 
arranged  in  coarse  and  fine  series  and  are  frequently  accompanied  by  dark  or 
bright  bands.  This  probably  also  accounts  for  the  effect  of  the  fore-and-aft 
motion  of  the  grating,  mentioned  above.  Moreover,  it  would  be  interesting 
to  search  for  repetitions  of  given  groups  of  fringes  while  the  grating  is  being 
shifted  parallel  to  itself,  from  end  to  end,  as  this  might  indicate  the  residual 
imperfections  of  the  screw  with  which  the  grating  was  ruled.  If  the  ocular  is 
drawn  and  set  outward  from  the  principal  focal  plane  (at  which  the  slit  image 
is  quite  sharp)  into  a  different  position,  the  fringes  move  in  a  direction  opposite 
to  the  grating.  If  the  ocular  is  set  inward  from  the  principal  focal  plane,  they 
move  in  the  same  direction  as  the  grating.  This  would  not  be  unexpected ;  but 
secondary  fringes  or  something  else  in  the  field  seem  to  remain  stationary. 
Successive  fields  may  be  quite  different  as  to  arrangement  of  fine  and  coarse 
lines,  but  all  plane  gratings  exhibit  the  same  phenomena.  Thus  it  is  obvious 
that  the  fringes  of  the  present  paper  result  from  a  residual  irregularity  in  the 
rulings  of  the  grating.  Micrometrically,  the  successive  strips  of  a  slit  image, 
however  fine,  are  of  unequal  intensity.  Between  these  there  is  diffraction,  as 
may  be  tested  by  examining  the  clear  glass  at  the  edge  of  the  ruled  space. 

To  attempt  a  theory  of  these  phenomena  seems  premature ;  but  it  is  obvious 
that  in  the  otherwise  indistinguishable  images  of  a  slit  in  homogeneous  light, 
however  sharp  or  however  narrow,  the  nature  of  its  origin  still  persists  and 
may  be  detected  by  observations  outside  of  the  principal  focal  plane.  A  fine 
slit  is  in  all  cases  presupposed,  and  all  the  phenomena  vanish  for  a  wide  slit. 
On  the  other  hand,  the  width  of  the  pencils  of  parallel  rays  may  be  far  greater 
than  is  necessary  to  show  the  strong  Fraunhofer  lines,  if  indeed  there  is  any 
limitation  to  this  width. 


CHAPTER  VII. 


PRISMATIC  LONG-DISTANCE  METHODS  IN  REVERSED  AND  NON-REVERSED 
SPECTRUM  INTERFEROMETRY. 

51.  Purpose.  —  It  is  preliminarily  the  object  of  the  present  paper  to  examine 
a  variety  of  new  methods  for  the  production  of  interferences  with  spectra, 
with  a  view  to  the  selection  of  as  simple  a  design  as  possible  for  practical  pur- 
poses.   Some  interesting  differences  appear  in  the  results,  so  that  the  sim- 
plicity of  construction  does  not  necessarily  recommend  the  apparatus  for  use. 

In  the  second  place,  the  endeavor  will  be  made  to  assemble  appurtenances 
in  such  a  way  that  the  extremely  mobile  phenomena  may  be  under  control, 
even  in  a  moderately  agitated  laboratory.  In  case  of  the  early  interferometer 
experiments,  the  interferences  disappeared  on  merely  touching  the  apparatus, 
and  are  rarely  or  never  at  rest;  whereas  it  is,  of  course,  necessary  that  they 
should  remain  visible  while  the  micrometer  is  being  moved.  These  experi- 
ments are  now  nearly  completed,  but  will  preferably  be  described  in  a  succeed- 
ing report. 

52.  Methods  and  apparatus.  —  Some  prismatic  methods  wefe  tested  in  the 
earlier  volume,  but  not  developed;  for  the  plan  of  using  a  transmitting  grating 
twice,  or  two  gratings  in  succession,  seemed  to  contain  greater  promise.    The 
prism  method  is,  however,  more  sim- 

ple than  any  of  the  others  and  there- 
fore deserving  of  special  study. 

In  figure  72  the  large  right-angled 
prism  P,  with  its  faces  silvered,  re- 
ceives  the  pencil  of  parallel  white 
rays,  L,  on  its  orthogonal  faces  and 
reflects  them  to  the  plane  opaque  _  o 


mirrors  n  and  m.  From  here  the  rays 
are  further  reflected,  either  nearly  in 
parallel,  as  in  the  figure,  or  crossed, 
as  at  c,  c',  to  the  remote  opaque 
mirrors  N  and  M,  which  in  turn  re- 
flect them  to  the  plane  or  concave 
grating  G.  If  the  rays  converge  at  the 
appropriate  angle  of  diffraction,  0,  a  ~ 

selected  color  will  be  diffracted  in  the  direction  of  the  normal  to  G  in  each 
case.  If  the  two  paths  are  nearly  equal,  these  rays  will  therefore  interfere 
in  the  axis  GT  and  the  results  may  be  observed  by  a  telescope  or  a  lens  at  T. 
In  my  apparatus  the  distances  mM  and  nN  were  of  the  order  of  2  meters. 
In  consequence  of  the  three  successive  reflections,  it  is  somewhat  difficult  to 
102 


c/lf 


REVERSED   AND   NON-REVERSED   SPECTRA.  103 

obtain  spectrum  lines  normal  to  the  axis  of  the  spectrum,  so  that  if  the  latter 
are  superposed  the  lines  will  be  at  an  angle.  But  if  this  is  small,  it  does  not 
seriously  interfere  with  the  occurrence  of  fringes,  as  they  extend  from  top 
to  bottom  of  the  spectrum. 

The  appearance  in  general  is  of  the  linear  character  heretofore  described. 
They  pass  symmetrically  from  extreme  fineness,  through  a  maximum  size,  to 
fineness  again,  with  the  fore-and-aft  motion  of  the  grating  G,  and  they  usually 
rotate  near  the  maximum. 

If  the  mirror  M  is  displaced  nearly  in  a  direction  normal  to  itself,  on  a 
micrometer,  the  fringes  undergo  the  same  evolution,  and  in  this  respect  differ 
from  the  case  where  the  primary  differentiator,  P,  was  also  a  grating.  In 
this  case  the  displacement  of  M  showed  no  discernible  modifications  of  the 
size  or  character  of  the  fringe  pattern.  The  fringes  merely  moved.  In  figure 
72  the  effect  of  moving  G  or  M  fore  and  aft  is  similar,  since  it  throws  the 
point  of  convergence  of  the  rays  NG  and  MG  in  front  of  or  behind  the  grating. 
The  result  is  therefore  different  when  white  light  impinges  on  G  from  what  it 
is  when  the  light  is  already  nearly  homogeneous. 

The  limit  of  visibility  is  also  inferior  to  the  double-grating  method  heretofore 
used,  for  the  fringes  passed  between  the  limits  of  visibility  through  the  maxi- 
mum size,  for  a  displacement  of  M  of  only  about  3  mm.  Smaller  ranges 
may  occur.  On  limiting  the  incident  beam  at  L  to  a  breadth  of  about  0.5  cm., 
the  fringes  became  much  broader  and  relatively  intense. 

There  is,  of  course,  an  abundance  of  light,  so  that  the  screening  of  the 
incident  beam  is  not  disadvantageous.  In  this  case,  when  the  fore-and-aft 
position  (illuminated  strips  on  the  grating  coincide,  as  in  figure  72)  and  the 
position  of  the  grating  relative  to  its  normal  axis  were  carefully  adjusted, 
large  arrow-headed  fringes,  as  in  figure  73,  were  obtained,  usually  less  closely 
packed  vertically.  Apart  from  tremors,  these  move  slowly  up  and  down 
(breathing),  as  a  result,  no  doubt,  of  changes  of  temperature  in  the  air- 
paths.  A  mica  film  inserted  into  one  beam  and  slowly  rotated  produced 
similar  motion,  besides  introducing  its  own  grid  of  vertical  and  parallel  fringes. 
The  reason  for  the  occurrence  of  these  arrows  is  not  quite  clear  to  me,  though 
they  are  associated  with  horizontal  fringes  and  homogeneous  light,  the  doubly 
inflected  forms  belonging  to  inclined  fringes  and  homogeneous  light. 

In  the  endeavor  to  reproduce  these  fringes  with  the  sodium  arc,  I  failed 
after  long  trials.  The  reason  may  be  sought  in  the  nicker  of  the  arc,  whereby 
the  beam  passes  from  one  side  to  the  other  of  the  edge  of  the  prism  P,  but  it 
is  probably  due  to  the  inadmissibility  of  a  wide  slit. 

53.  The  same.  Crossed  rays. — The  present  method,  using  four  mirrors, 
has,  nevertheless,  the  advantage  of  admitting  the  use  of  either  parallel  or 
crossed  rays.  Inasmuch  as  these  rays  are  white  until  they  leave  the  grating, 
the  method  is  interesting.  On  being  tested  it  showed  the  same  peculiarities 
as  the  preceding.  The  crossed  rays  (ccf,  figure  72)  are  more  nearly  normal 
to  the  mirrors  M  and  N;  nevertheless  the  range  within  which  the  interfer- 


104 


THE   INTERFEROMETRY   OF 


ences  are  visible  is  not  above  2  mm.  of  displacement  of  M.  The  fringes  may, 
as  usual,  be  made  as  large  as  possible,  by  first  superposing  the  two  illuminated 
strips  on  the  grating  G  (by  fore-and-aft  motion)  and  then  rotating  the  grating 
on  an  axis  normal  to  its  face  until  the  best  conditions  appear.  Both  spectra 
are  very  bright,  but  liable  to  be  in  different  focal  planes  from  inadequate 
planeness  of  the  reflecting  system.  If  work  of  precision  is  aimed  at,  this  con- 
dition is  of  foremost  importance. 

54.  Another  method. — If  the  opportunity  of  using  crossed  pencils  of  white 
light  is  to  be  dispensed  with,  the  prism  method  may  be  simplified,  as  shown 
in  figure  74.  Here  P  is  a  prism  with  silvered  sides  and  a  prism  angle  of  less 
than  30°.  It  receives  horizontal  white  rays  L  from  a  collimator,  which,  after 
reflection  from  the  opaque  mir- 
rors M  and  N,  impinge  on  the 
grating  G,  plane  or  concave,  and 
are  observed  at  T  by  a  telescope 
or  lens. 

If  <p  is  the  prism  angle  and  0 
the  angle  of  diffraction,  it  is 
easily  seen  that  the  angle  be- 
tween the  rays  reflected  at  M 
or  N  is 


74 


Hence,  if  P  is  a  30°  prism,  the 

observations  can  be  made  only 

in  the  second-order  spectra.    If  cJlf 

observations  in  the  first  order 

are  desired  because  of  the  greater  illumination,  <p  must  be  less  than  20°,  as  a 

rule,  for  a  grating  of  about  15,000  lines  to  the  inch.    The  mirrors  M  and  N 

make  an  angle  of  <7/2  =  O+0)/2  with  the  line  MN. 

The  first  experiments  were  made  with  a  30°  prism  and  second-order  spectra 
from  a  concave  grating  (Z}  =  i  77X10-*  cm.).  Sunlight  was  used.  The  two 
superposed  spectra  were  magnificent,  with  abundance  of  light  and  high  disper- 
sion; but  the  spectra  were  of  unequal  intensity  and  in  different  focal  planes, 
so  much  so  that  the  images  of  the  guiding  horizontal  thread  of  the  spectra 
could  scarcely  be  seen  together.  This  made  the  adjustment  for  coincident 
longitudinal  axes  very  difficult,  and  the  interferences  were  not  found  until  after 
long  trial.  The  reason  for  this  is  the  probable  concavity  or  convexity  of  one 
or  more  of  the  reflecting  surfaces.  Another  difficulty  was  the  distance  apart 
of  the  mirrors  M  and  N  (roughly,  150  cm.  for  a  distance  of  about  2  meters 
from  P  to  T),  so  that  it  was  inconvenient  to  observe  and  actuate  the  mirror 
micrometer  at  M.  Further  attempt  at  improvement  was  therefore  abandoned. 

This  prism  was  now  replaced  by  one  of  less  angle  than  ^  =  20°,  also  well 
silvered.  In  the  first  experiments  the  adjustment  did  not  admit  of  a  coinci- 
dence of  light,  except  near  the  C  line  of  the  red;  but  M  and  AT  were  now  less 
than  90  cm.  apart,  while  the  distance  between  G  and  T  was  about  1 10  cm.,  and 


REVERSED   AND   NON-REVERSED   SPECTRA. 


105 


between  G  and  P  about  10  cm.  In  this  case  the  focal  planes  were  nearly  iden- 
tical and  the  interferences  easily  found  in  the  red  region  between  the  two  C 
lines.  They  appeared  as  small  red  pearls,  very  vivid  on  limiting  the  lateral 
extent  of  the  pencil  L  to  about  5  mm.,  but,  to  my  astonishment,  they  very 
soon  vanished  on  displacing  M  in  a  direction  normal  to  itself  i  or  2  mm. 

55.  Methods  using  prismatic  dispersion. — The  small  range  of  displace- 
ment available  in  the  prismatic  reflection  methods  induced  me  to  devise 
corresponding  refraction  methods,  to  see  whether  these  would  show  any 
advantage  in  this  respect.  Accordingly  the  interferometer  (fig.  75)  was  in- 
stalled and  the  fringes  found  without  much  difficulty.  Here  P  is  the  symmet- 
rical prism,  receiving  the  collimated  beam  of  incident  white  light  on  the  faces 
meeting  at  the  obtuse  edge  and  refracting  them  in  relation  to  the  smaller 
prism  angle  <p.  This  must  be  less  than  45°,  for  convenience  in  observation, 
as  otherwise  the  dispersed  beams  meeting  the  opaque  mirrors  M  and  N  will 


76 


be  too  far  apart  for  manipulation,  supposing,  of  course,  that  the  distance 
PM  and  PN  are  over  a  meter.  I  used  an  equilateral  90°  prism  for  want  of  a 
better.  The  spectra  reflected  from  M  and  N  respectively  impinge  on  the 
grating  G,  concave  or  plane,  and  are  viewed  at  T  with  a  lens  or  telescope. 
In  consequence  of  the  large  angle  0,  second-order  spectra  were  used,  without 
apparent  disadvantage.  The  dispersion  of  P  and  G  being  summational,  the 
total  is  very  large. 

To  return  to  the  angles  again,  if  0  denotes  the  obtuse  prism  angle,  and  r  the 
angle  of  refraction,  the  angle  of  incidence  is  90°  —  </>/2,  or 

(1)  cos  0/2  =M  sin  r 
Again, 

(2)  sin  i'  =  n  cos  (0/2  -f-r) 
when  *'  is  the  angle  of  emergence.    Hence 

sin  i'  =  cos  0/2  (\/M2  —  cos2  0/2  —  sin 


106  REVERSED  AND   NON-REVERSED   SPECTRA. 


-i  — 0-  Thus  if  /*=  i.55>tnensin*'  = 
and  ^'  =  28.4°.  Now,  since  i+8  =  6,  the  angle  6  will  obviously  have  to  be  in 
the  second  order  of  the  spectra  of  the  grating  G. 

Although  the  two  spectra  obtained  in  this  way  were  highly  dispersed  and 
very  brilliant,  the  interference  phenomenon  itself  was  not  much  superior  to 
the  case  where  reflection  from  the  (silvered)  faces  of  the  prism  was  employed. 
The  fringes  disappeared,  in  fact,  for  a  displacement  of  i  or  2  mm.  of  the  mirror 
M,  showing  the  usual  inflation  of  form  just  before  vanishing.  The  details 
also  were  of  the  same  nature,  the  large  arrow-shaped  forms  being  obtained 
when  illuminated  strips  on  the  grating  were  superposed  and  the  latter  slightly 
rotated  until  the  maximal  conditions  appeared. 

To  increase  the  range,  the  angle  8  must  be  reduced,  as  far  as  practicable. 
This  is  possible  in  the  present  method,  since  the  points  of  intersection  at  a 
and  G  may  be  made  to  all  but  coincide.  Reflection  from  the  mirrors  M  and 
N  would  then  be  normal.  To  attain  this  end  it  will  be  necessary  either  to 
have  the  grating  constant  or  the  prism  angle  <f>  predetermined,  or  to  use  rays 
of  suitable  divergence  at  L. 

56.  Methods  with  paired  prisms.— White  light  (fig.  76,  L)  from  a  collimator 
is  reflected  in  turn  from  the  silvered  sides  of  the  sharp  prism  P,  from  the 
opaque  mirrors  M  and  N,  and  from  the  silvered  blunt  prism  P',  as  shown  by 
the  component  beams  abc  and  a'b'c' .  Thereafter  the  white  beams  are  diffracted 
by  an  Ives  film  grating  G,  with  attached  prism  p,  and  observed  in  a  telescope 
at  T.  Interference,  therefore,  takes  place  in  the  focal  plane  of  the  telescope 
and  would  not  (for  the  case  in  fig.  76)  occur  in  its  absence.  Very  interesting 
results  were  obtained  with  this  apparatus.  The  spectra  are  non-reversed  or 
else  (if  slit  and  grating  are  rotated  90°)  inverted.  The  work,  however,  is 
still  in  progress  and  will  be  described  elsewhere.  I  will  merely  add,  in  this 
place,  that  the  work  with  prisms  is  important,  inasmuch  as  it  shows  the  essen- 
tial part  played  by  the  diffraction  of  the  slit  of  the  collimator,  in  its  bearing 
on  the  phenomena  of  the  present  report.  It  is  the  function  of  the  prism  P  to 
cleave  the  diffracted  field  which  leaves  the  collimator.  For  this  reason  pencils 
identical  in  source  are  found  on  both  sides  of  P.  The  experiments  thus  fur- 
nish the  final  link  in  the  theory  of  the  phenomena. 

Furthermore,  as  the  above  results  already  show,  the  range  of  displace- 
ment of  either  opaque  mirror  (M,  N)  within  which  interference  fringes  are 
visible,  increases  in  marked  degree  with  the  dispersion  to  which  the  white  ray 
is  subjected  on  separation  and  before  the  resulting  partial  rays  reach  their 
final  recombination.  These  ranges  increase  from  a  fraction  of  a  millimeter 
to  almost  a  centimeter,  while  the  width  of  the  strip  of  spectrum  carrying 
the  interference  fringes,  caet.  par.,  remains  the  same.  This  also  has  a  fun- 
damental bearing  on  the  phenomenon  and  is  under  investigation.  The  ques- 
tion at  issue  is  whether  increase  of  range  of  displacement  results  simply 
from  the  geometry  of  the  optic  system,  or  whether  wave-trains  are  actually 
uniform  throughout  greater  lengths,  in  proportion  as  they  have  been  more 
highly  dispersed. 


CHAPTER  VIII. 


THE  LINEAR  TYPE  OF  DISPLACEMENT  INTERFEROMETERS. 

57.  Introductory. — This  apparatus  will  be  referred  to  in  various  places  in 
this  book  and  presents  certain  interesting  features.     The  incidence  of  the 
grating  is  normal  (I  =  R  =  o),  and  both  component  rays  in  their  vertical  pro- 
jection lie  strictly  in  the  same  plane.    To  make  the  horizontal  projection  also 
collinear  is  not  quite  possible  in  practice,  because  the  direct  or  unreflected 
rays  and  the  corresponding  spectra  would  overlap  with  the  spectra  of  the 
interferometer.    As  the  former  are  much  more  intense,  the  interference  pat- 
terns would  scarcely  be  visible  in  the  combination.    To  avoid  this,  the  rays 
diverge  slightly  (a  few  degrees,  depending  on  the  distance  between  grating 
and  opaque  mirrors)  in  a  vertical  plane.    But  this  is  of  no  consequence,  as  the 
horizontal  projections  only  are  used  in  the  measurements.    One  may  note, 
in  passing,  that  this  avoidance  of  coincidence  with  undesirable  spectra  secured 
by  tilting  the  grating  and  the  corresponding  opaque  mirror  in  the  same 
direction  is,  in  general,  one  of  the  essentials  of  the  adjustments. 

The  advantage  of  the  linear  displacement  interferometer  is  this:  that  it 
can  be  built  on  a  rail  and  mounted  along  a  wall  or  a  pier.  If  the  rail  is  tubular, 
a  current  of  water  may  be  passed  through  it  from  the  middle  toward  both 
ends,  to  insure  constancy  of  temperature. 

58.  Apparatus. — The  apparatus  was  constructed  as  follows  and  gave  good 
results  at  once,  showing  strong  interferences.     The  ellipses  were,  in  fact, 
oblate  in  the  red,  circular  in  the  yellow,  and  prolate  in  the  blues,  but  clear 
throughout. 


77 


Light  enters  from  an  arc  lamp,  A,  or  Nernst  burner,  or  the  sun,  at  the  slit 
S,  and  is  collimated  by  the  lens  L.  Then  the  parallel  rays  pass  the  grating  G 
with  its  ruled  side  toward  L.  From  the  grating  the  reflected  beam  returns 
to  the  opaque  mirror  N,  and  is  then  reflected  into  the  auxiliary  or  adjustment 
telescope,  T.  The  component  beam  transmitted  at  G  is  reflected  from  the 
opaque  mirror  M,  returned  to  the  ruled  side  of  G,  and  thus  also  reflected 
into  T  coincidently  with  the  other  beam. 

107 


108  THE   INTERFEROMETRY  OF 

Figure  77  shows  that  the  entering  undivided  beam  LG  passes  just 
above  the  mirror  M,  and  is  reflected  just  below  this  from  the  top  of  N. 
Similarly,  the  reunited  beam  GT  passes  just  above  M,  but  is  reflected  from 
the  top  of  M ,  the  object  being  to  make  the  vertical  angle  at  G  as  small 
as  possible. 

The  mirrors  M  and  N  and  the  grating  G  are  on  adjustable  bases,  a,  a',  a", 
each  controlled  by  three  leveling  screws  on  a  plane-dot-slot  arrangement  in 
the  tablets  m,  m',  m",  the  axis  of  rotation  being  horizontal  and  normal  to  the 
diagram.  The  tablets,  furthermore,  may  be  revolved  and  raised  or  lowered 
by  the  rods  n,  n',  n",  which  are  attached  by  ordinary  clamps  to  the  large, 
tubular,  horizontal  rail,  RR,  in  question,  admitting  of  a  circuit  of  water.  The 
latter  is  secured  to  the  pier. 

The  angles  of  inclination  of  the  figure  are  much  exaggerated,  since  the  dis- 
tance MG=GN  (nearly)  is  from  one-half  to  several  meters  in  extent. 

The  mirror  M  is  on  a  Fraunhofer  micrometer  suggested  at  m.  The  bases, 
a,  a',  a",  are  drawn  to  the  tablets,  m,  mf,  m",  by  firm  springs,  preferably  run- 
ning into  the  tubes  below  them. 

The  axis  of  the  adjustment  telescope,  T,  lies  in  the  plane  of  the  figure  and 
serves  the  purpose  of  bringing  the  direct  slit  images  into  horizontal  and  vertical 
coincidence.  When  this  is  done  it  may  be  removed,  if  desirable,  as  the  ray 
GT  is  not  thereafter  used.  T  should  not  be  attached  to  the  rail,  but  placed 
on  an  independent  table,  or  standard,  so  as  not  to  be  an  integrant  part  of  the 
interferometer.  The  telescope,  T  (not  shown),  for  the  observation  of  the 
interferences,  should  be  independently  mounted  on  the  same  table.  This  tele- 
scope lies  outside  of  the  diagram,  to  the  right  or  the  left  of  it,  to  catch  either 
of  the  two  diffraction  spectra  selected.  It  will  be  seen  that  these  lie  quite 
above  the  direct  diffraction  spectra  of  the  ray  LGM.  Otherwise,  as  this  is 
much  more  intense,  it  would  completely  wipe  out  the  interference  spectra 
and  their  combination.  The  latter,  when  seen  alone,  are  very  brilliant,  black 
and  colored  patterns,  running  through  the  spectrum  when  the  micrometer, 
m,  is  manipulated.  If  the  distance  GN  is  large  and  the  grating  G,  as  usual, 
slightly  wedge-shaped,  the  superfluous  rear  reflection  from  G  may  be  blotted 
out  at  N  by  a  small  screen.  It  is  easily  recognized,  as  it  is  brown  from 
scattered  light. 

The  installation  is  simple.  The  parts  being  adjusted  nearly  symmetrically, 
the  undivided  ray  from  a  wide  slit  is  brought  to  the  top  of  M  by  raising  or 
lowering  the  lamp.  This  should  first  be  done  roughly  with  the  lens  and  slit 
removed.  N  has  at  the  same  time  been  placed  just  below  the  beam,  and  this 
passes  through  the  middle  part  of  G.  The  latter  is  then  inclined  by  the  adjust- 
ment screws  until  the  component  beam  GN  strikes  the  top  of  N,  symmetri- 
cally. Next  N  is  inclined  and  rotated  (vertical  axis)  until  the  reflected  beam 
enters  the  telescope,  T.  Finally,  M  is  inclined  and  rotated  (vertical  axis) 
until  the  reflected  rays  MG  and  GT  also  enter  the  telescope,  the  final  sharp 
adjustment  being  made  with  a  narrow  slit  and  the  eye  at  the  telescope. 
The  mirror  M  must  also  have  a  fine  vertical  adjustment  (not  shown).  If  the 


REVERSED   AND   NON-REVERSED   SPECTRA.  109 

distances  NG  (face  toward  the  light)  and  MG  are  equal,  the  interferences 
are  then  easily  found  by  moving  the  mirror  M  on  the  micrometer  toward 
the  grating. 

As  compared  with  the  other  non-linear  interferometers  used  under  like 
conditions,  the  present  instrument,  even  when  mounted  on  a  K-inch  gas- 
pipe,  RR,  showed  itself  remarkably  steady,  so  that  rings  could  be  observed 
in  spite  of  the  tremors  of  the  hill  on  which  the  laboratory  is  built. 

59.  Film=grating  adjustment.  Michelson's  interferometer.— If  the  grat- 
ing G  is  a  film  grating,  like  those  in  the  market,  with  14,000  lines  to  the  inch, 
it  should  be  mounted  smoothly  on  the  unruled  side,  on  a  thick  glass  plate, 
with  Canada  balsam,  and  without  a  cover  plate  for  the  ruled  side.  It  is  to  be 
adjusted  with  the  glass  side  toward  the  source  of  light,  so  that  the  reflection 
taken  may  be  from  this  side  only  (see  r,  fig.  78).  In  the  telescope,  T,  directed 
toward  the  reflected  beams,  two  slits  (one  for  each  component  beam)  only 
appear,  as  the  glass  plate  does  not  reflect  on  the  side  covered  by  the  grating 
(g  in  fig.  78).  The  slits  placed  in  coincidence  will  then  show  the  elliptic  inter- 


Ic/Jt 


ferences  in  the  diffracted  beam  D  at  the  proper  distances.  With  so  large  a 
dispersion  as  the  above,  the  ellipses  are  usually  too  large.  They  should  then 
be  reduced  in  size  by  a  compensator  placed  in  the  beam  on  the  ruled  side  of 
the  grating;  or,  preferably,  the  grating  may  be  mounted  on  a  plate  of  glass 
fully  i  cm.  (or  more)  thick,  as  in  figure  78.  This  thick  plate  has  the  additional 
advantage  of  eliminating  the  stationary  interferences  due  to  the  front  and 
rear  faces  of  the  grating.  In  case  of  thin  glass  plates  (2  or  3  mm.),  these 
stationary  interferences  are  very  strong,  coarse,  vertical  lines  and  exceed- 
ingly annoying. 

If  the  film  grating  is  carefully  mounted  in  this  way,  it  is  nearly  as  good  as 
a  ruled  grating.  There  is,  however,  one  insuperable  objection,  inasmuch  as 
the  ruled  face,  though  it  does  not  reflect  sharply,  does  diffract,  and  this  more 
strongly  than  the  other.  Thus  there  are  always  3  superposed  spectra  in 
the  telescope,  the  third  coming  from  the  film  side  only,  whereas  the  other  two 
are  produced  by  the  rays  coming  coincidently  from  r  on  the  unruled  side  of 
the  grating.  Hence  the  velvety  blackness  of  the  interferences  in  case  of  the 
ruled  gratings  can  not  be  reproduced  by  the  film  grating,  since  the  interfer- 
ences are  spread  out  on  a  colored  ground.  They  are,  however,  quite  strong 
enough  for  all  practical  purposes,  and  the  lines  are  sharply  and  symmetrically 


HQ  THE   INTERFEROMETRY  OF 

traced.  A  vertical  wire  2  or  3  mm.  thick,  placed  symmetrically  in  front  of 
the  objective  of  the  telescope,  makes  the  interference  relatively  strong  and 
sharp,  by  blotting  out  the  third  spectrum  partially;  but  it  at  the  same  time 
diminishes  the  light  available.  A  wide  slit  in  front  of  the  objective  subserves 
the  same  purposes  better. 

If  the  distance  apart  of  the  mirrors  M  and  N  and  the  grating  G  is  large,  it 
is  best  to  dispense  with  the  rail  RR  altogether,  and  to  mount  the  mirrors  and 
grating  directly  on  the  pier  or  wall.  This  has  the  additional  advantage  of  a 
large  free  space  between  M  and  G  or  G  and  N,  so  that  spacious  apparatus  like 
a  fog-chamber  may  be  independently  mounted  there.  This  was  the  case  in 
the  optic  experiments  on  the  thermal  coefficients  of  the  refraction  of  air,  etc., 
below,  where  the  distance  between  MG  and  GN  was  nearly  2  meters.  In  such 
a  case,  moreover,  in  addition  to  the  usual  three  adjustment  screws  of  the 
mirror  M  at  the  micrometer,  it  is  desirable  to  have  two  others  bearing  on  the 
rigid  parts  of  the  support,  so  that  the  final  adjustment  may  be  made  elastically. 
By  devising  a  tetrahedral  plan  of  bracing  M,  G,  N,  independent  of  each  other, 
using  short  rods  and  clamping  all  parts  on  relatively  short  stems,  I  eventually 
obtained  a  mounting  which  was  almost  free  from  tremors,  even  amid  the  dis- 
turbances of  the  surrounding  laboratory.  In  figure  79  one  of  these  mount- 
ings is  suggested :  a  and  b  are  y^-inch  gas-pipes  (about  a  foot  long) ,  sunk  into 
the  wall  of  the  pier  at  their  rear  ends ;  cd  is  a  cross-rod  of  same  size  and  material, 
damped  in  place,  and  supporting  the  grating  (or  a  micrometer)  G.  The 
screw  h  abutting  against  the  wall  gives  the  horizontal  elastic  adjustment. 
The  braces  e  and  /,  which  may  be  adjusted  by  rotation  (screw)  abutting  in  g 
at  the  wall,  give  the  grating  vertical  elastic  adjustment.  Thus  h,  e,  and  /, 
rotate  G  around  vertical  and  horizontal  axes,  respectively. 

60.  Michelson's  interferences.— If  the  collimator,  SL,  is  removed  and 
replaced  by  a  strong  sodium  flame  provided  with  a  condenser,  Michelson's 
interferences  will  appear  at  T  when  the  instrument  is  in  adjustment.  It  is 
rather  surprising  that,  even  in  case  of  a  film  grating  adjusted  as  above,  they 
are  well-defined  circles  covering  the  whole  field  of  the  telescope.  If  the  col- 
limator SL  is  retained  and  the  sodium  light  introduced  from  the  side  by  aid 
of  a  reflecting  mirror,  placed  between  the  grating  G  and  the  collimating  lens 
L,  both  interferences  may  be  observed  at  the  same  time  in  corresponding 
telescopes.  The  mirror  introducing  the  homogeneous  light  should  in  such  a 
case  be  provided  with  a  clear  space  (silver  removed),  through  which  the  white 
beam,  SL,  may  pass  without  obstruction.  In  a  vertical  plane  the  interferences 
have  the  same  size  and  character  at  the  sodium  line.  Horizontally  the  spec- 
trum interferences  vary  with  the  dispersion. 

If  an  apparatus  constructed  of  gas-pipe  is  employed,  however,  it  is  far  too 
frail  for  the  practical  use  of  the  Michelson  interferences.  Vibrations  within 
the  apparatus  are  excited  on  merely  touching  it.  For  the  purpose  of  displace- 
ment interferometry,  however,  such  an  apparatus  is  quite  adequate;  for  the 
measurements  are  taken  when  the  tremors  have  vanished. 


REVERSED   AND   NON-REVERSED   SPECTRA.  Ill 

61.  Film  grating.    Another  adjustment.  —  The  supernumerary  spectra  may 
be  gotten  rid  of  altogether  by  using  the  method  shown  in  figure  80.    Here  the 
impinging  vertical  sheet  of  white  light,  L,  from  the  collimator,  falls  upon  the 
clear  or  unruled  part  p  of  the  plate  of  the  grating,  the  film  extending  out  as 
far  as  shown  at  G.  If  M  and  N  are  the  opaque  mirrors,  the  reflected  rays  a  and 
b  passing  G  are  additionally  reflected  into  a'  and  b',  and  thence,  after  leaving 
the  grating,  into  c  and  d.    As  both  of  the  latter  pass  through  the  film,  both 
produce  spectra;  but  b'  and  e  may  be  blotted  out  by  a  screen  at  the  mirror 
N.    This  leaves  only  d  beyond  the  grating.    Again,  the  transmitted  ray  from 
L,  after  reflection  at  M,  is  again  reflected  into  c  and  d',  which  is  made  coinci- 
dent with  d  .    But  c,  being  reflected  from  the  unruled  side,  has  no  spectrum. 
Thus  the  spectra  due  to  the  two  rays  d  alone  interfere. 

Had  the  grating  been  reversed,  caet.  par.,  then  the  ray  c  would  have  pro- 
duced the  strongest  spectrum,  and  superposed  on  the  other  two  it  would 
have  greatly  diminished  the  clearness. 

In  the  telescope,  whereas  the  ray  a'  prolonged  is  white,  the  ray  d'  from  M 
and  reflected  from  the  film  is  strongly  azure  blue,  due  to  regularly  scattered 
light.  This  blue  image  is  apt  to  be  less  sharp,  unless  very  flat  parts  of  the 
film  are  found.  The  two  spectra,  however,  are  good  and  the  interferences 
satisfactory.  The  sodium  line  is  sufficiently  indicated,  though,  like  the  blue 
image,  not  quite  sharp. 

This  method  of  using  the  unruled  edge  of  the  plate  of  the  grating  for  reflec- 
tion is,  of  course,  equally  applicable  and  advantageous  in  the  case  of  the 
ruled  grating.  Only  the  two  interfering  spectra  and  no  diffused  light  are 
present  in  the  field  of  the  telescope,  and  if  sunlight  is  used  the  Fraunhofer 
lines  are  beautifully  sharp. 

62.  Equations.  —  The  equations  for  Ne/e,  for  normal  incidence  I=R  =  o, 
takes  its  simplest  form  as 

(i)  Nc/e  =  M  -  Un/d\  =  A  +35/X2,  nearly 

where  Nc  is  the  coordinate  of  the  center  of  a  given  ellipse  on  the  micrometer 
M,  for  the  thickness  of  glass  grating  e,  index  of  refraction  n,  and  color  of  wave- 
length X. 

Hence  if  two  different  wave-lengths,  X  and  X',  are  in  question  (5  refers  to 
differences), 

w 

8N,  being  the  displacement  of  the  micrometer  to  pass  the  center  of  ellipses 
from  line  X  to  line  X'. 


If  M=A+£/X2  and  Xd/i/dX=  -2#/X2,  then 
(3)  «tf.-3 


112  REVERSED   AND   NON-REVERSED   SPECTRA. 

from  which  B  may  be  obtained  without  further  measurements.  If  greater 
approximation  is  necessary,  so  that  two  constants,  B  and  C,  enter  the  disper- 
sion equation, 

(4)  W<  =  3 


so  that  observations  at  three  spectrum  lines,  X,  X',  X",  would  be  necessary. 

The  amount  of  displacement  corresponding  to  the  thickness  e  of  glass  is, 
at  a  given  spectrum  line  X, 


where  2J5/X2  is  constant  for  all  values  of  e,  or 


It  is  therefore  not  possible  to  obviate  the  term  in  B,  determined  as  shown, 
if  n  is  to  be  measured. 

If  equal  distances  are  cut  off  at  M  and  N,  the  interference  pattern,  of  course, 
remains  stationary  in  the  spectrum.  It  is  interesting  to  inquire  to  what  degree 
this  may  be  guaranteed.  Equation  (3)  is  available  for  the  purpose,  and,  since 
X  and  X'  are  nearly  the  same,  X'—  X  =  SX  and 

-,,         ,->2<iX     6eB 


Let  5X  be  the  width  of  the  sodium  lines: 

5X  =  6Xio-"cm.  X  =  59Xicr6cm.  e  =  o.68cm.  B  =  4.6Xio~n 

data  for  the  above  grating  and  sodium  light.    Hence 
.„      6Xo.68X4-6Xio-11X6Xio-8 
8N*=  (59)'Xicr»  -=5-5Xio-  cm. 

i.e.,  about  a  half  of  iQ-4  cm.  This  would  be  equivalent  to  the  space  on  a  grating 
with  about  20,000  lines  to  the  centimeter,  or  50,000  to  the  inch.  The  ellipses 
can  not  be  set  as  closely  as  this,  but  the  order  of  sensitiveness  is  within  that 
of  a  good  micrometer. 

It  is  interesting  to  inquire  whether  the  sensitiveness  will  change  markedly 
for  larger  angles  of  incidence  I.  If  p.  is  the  index  of  refraction,  the  largest 
angle  R  obtainable  at  grazing  incidence,  1  =  90°,  would  be  sin  R=I/H.  It 
may  then  be  shown  that 


d\        X3  VM^ 

Putting  M=  1.5  and  the  other  data  as  above,  where  d\  =  6X  iQ-8  cm., 

dNf  =  L34><4^>0o^_  9.2  X  io-u/(59)2X  io-"+4.5 

(59)3Xio-»  !.I2  =  7-7Xio-5cm. 

The  datum  is  of  the  same  order  as  above,  so  that  the  sensitiveness  changes 
but  very  little  for  different  angles  of  incidence.  Thus  there  is  no  disadvan- 
tage in  using  7  =  o. 


CHAPTER  IX. 


THE  USE  OF  COMPENSATORS,  BOUNDED  BY  CURVED  SURFACES,  IN 
DISPLACEMENT  INTERFEROMETRY. 

63.  Introduction. — The  method  of  increasing  the  sensitiveness  of  the  dis- 
placement interferometer  by  increasing  the  dispersion  of  the  grating  readily 
suggests  itself,  but  unfortunately  the  interference  pattern  loses  sharpness  in 
the  same  ratio  and  ultimately  becomes  too  diffuse  for  practical  purposes. 
Similar  sensitiveness  is  secured  when  the  air-paths  and  the  glass-paths  of  the 
component  beams  of  light  are  respectively  identical,  with  the  same  inadequacy 
in  the  huge  mobile  figures,  for  the  purpose  of  adjustment.    In  fact,  if  for  sim- 
plicity we  consider  the  incidence  normal  (I  =  R  =  o,  linear  interferometer), 
the  sensitiveness  becomes 

de/dn  =  \*/[*eD  cos  0.  ((n+2b/\*)-N))] 

where  6  is  the  angle  of  diffraction  for  the  wave-length  \,  e  the  thickness  of  the 
plate  of  the  grating,  n  its  index  of  refraction,  D  the  grating  space,  n  the  order 
of  the  fringe,  and  b,  N,  constants.  Hence,  other  things  being  equal,  dQ/dn 
increases  as  D  and  e  grow  smaller,  where  e  =  o  is  obtained  by  a  compensator 
counteracting  the  thickness  of  the  plate  of  the  grating. 

It  occurred  to  me  that  the  difficulty  of  diffuse  interference  patterns  might 
be  overcome,  in  part,  by  the  use  of  compensators  with  curved  faces,  when  the 
case  would  become  similar  to  the  conversion  of  the  usual  interference  colors 
of  thin  plates  into  Newton's  rings.  Naturally  a  cylindric  lens  with  its  elements 
normal  to  the  slit  is  chiefly  in  question,  though  an  ordinary  lens  also  presents 
cases  of  interest,  chiefly  because  of  the  easy  conversion  of  elliptic  into  hyper- 
bolic patterns,  and  the  lens  is  more  easily  obtained. 

Other  methods  were  tried.  For  instance,  on  using  a  Fresnel  biprism  with 
its  blunt  edge  normal  to  the  slit,  two  sets  of  interference  patterns,  one  above 
the  other  in  the  spectrum,  are  obtained.  When  the  blunt  edge  is  parallel  to 
the  slit,  either  side  of  the  prism  gives  its  own  interferences,  but  they  can 
not  be  made  clearly  visible  at  the  same  time.  A  doubly  reflecting  plate  or  a 
thin  sheet  of  mica  covering  one  half  of  the  beam  will  produce  two  intersecting 
patterns,  but  these  also  are  of  little  use  for  measurement. 

64.  Lens  systems. — If  but  a  single  compensator  is  to  be  used,  i.e.,  compen- 
sation in  one  of  the  component  beams  only,  the  lens  in  question  must  be  of 
very  small  focal  power;  otherwise  the  adjustment  will  be  impossible,  as  the 
two  direct  images  of  the  slit  will  be  in  very  different  focal  planes.    Moreover, 
the  focal  power  should  be  variable.    All  this  makes  it  necessary  to  use  a 
doublet,  preferably  consisting  of  lenses  of  the  same  focal  power,  respectively 
convex  and  concave.    If  these  lenses  are  themselves  weak,  say  i  meter  in 
focal  distance,  both  slit  images  may  easily  be  seen  in  the  telescope  and  be 

113 


114 


THE   INTERFEROMETRY   OF 


sufficiently  sharp  for  adjustment.  If  the  lens  first  struck  by  light  is  convex 
and  the  second  concave,  their  focal  distances /i  and/2,  respectively,  and  their 
distances  apart  D,  the  focal  power  i/F  of  the  combination  used  is 

(i)  D/M-D/J* 

since  /i  =/2  =/.  The  position  of  the  equivalent  lens  is  d = DF/fi  =/2  =/.  D,  d 
are  both  measured  from  the  second  or  concave  lens  to  the  convex  lens,  and  D 
would  always  be  smaller  than  /.  If  the  lens  system  is  reversed,  F  remains 
the  same  as  before  for  the  same  D,  the  system  being  again  convex,  but  d  is 
reversed.  The  equivalent  lens  again  lies  toward  the  convex  side  of  the  system. 
In  other  words,  the  equivalent  lens  generally  lies  on  the  same  side  of  the 
doublet  as  the  convex  lens. 

In  the  actual  experiment,  however,  the  rays  go  through  the  lens  system 
twice.  In  this  case  it  is  perhaps  best  to  compute  the  distances  directly.  Of 
the  two  adjustments,  the  one  with  the  concave  lens  toward  the  grating  and 
the  convex  lens  toward  the  mirror  has  much  the  greater  range  of  focus  relative 
to  the  displacement  D.  Supposing  the  mirror  appreciably  in  contact  with  a 
convex  lens,  therefore,  if  b  is  its  principal  focal  distance  measured  from  the 
concave  lens,  b-\-D  =  M  its  principal  focal  distance  from  the  convex  lens  or 
mirror, 


where  fi  is  the  (numerical)  focal  distance  of  the  concave  and  ft  that  of  the 
convex  lens.    If  we  now  write 

(3)  b 

equation  (2)  is  easily  converted  into 

(4) 


--  =  - 
2B~f2 


so  that  the  usual  value  of  the  principal  focal  distance  has  been  halved  rela- 
tively to  the  new  position  of  the  equivalent  lens.    If,  as  in  the  present  case, 


f     /2_ 


2D(f+D) 

The  following  table  shows  roughly  the  corresponding  values  of  D  and  M  in 
centimeters : 


D 

M=C+D 

2B 

d 

2 

2450 

2500 

49 

5 

950 

IOOO 

47 

10 

455 

500 

45 

15 

292 

333 

4i 

20 

25 

212 
165 

250 

200 

38 
35 

REVERSED   AND   NON-REVERSED   SPECTRA. 


115 


As  6  is  smaller  than  B  by  equation  (3),  the  equivalent  lens  is  on  the  side  of  the 
convex  lens  and  at  a  distance 


behind  the  mirror,  or 

B-b=f(f+2D)/2(f+D) 
behind  the  concave  lens. 

If  the  system  is  reversed, /i  and  fz  are  to  be  replaced  by  — /i  and  —  ft,  whereas 
D  remains  positive.    Thus  the  equations  become  successively 

i=       !/(/!-£>) -2//2        [  T 


If /i  =/•*=/,  then 


2D    J-D 


Hence  the  equivalent  lens  has  the  same  focal  distance  as  before,  but  it  is 
now  placed  in  front  of  the  system,  at  a  greater  distance  than  it  was  formerly 
behind  it.  Measured  from  the  mirror  (mirror  distances,  M)  the  data  (in 
millimeters)  are  roughly  as  follows  : 


D 

2B 

B-  M 

2 

2500 

-5i 

5 

1000 

-52 

10 

500 

-54 

15 

333 

-56 

20 

250 

-57 

25 

200 

-58 

The  total  displacement  of  the  equivalent  lens  on  reversal  is  about  i  meter, 
falling  off  to  96  cm.  in  the  extreme  case.  The  image  is  larger  if  the  convex 
lens  is  nearer  the  grating  and  the  concave  lens  nearer  the  mirror. 

65.  Effective  thickness  of  the  lenticular  compensator. — The  compensator 
with  curved  faces  may  change  the  interference  pattern  in  two  ways;  viz,  by 
changing  the  angle  of  incidence  and  refraction  of  the  rays  at  the  grating,  and 
by  changing  the  path-difference  of  successive  rays  passing  through  it.  Both 
conditions  are  virtually  the  same,  or  at  least  occur  simultaneously.  If  there 
is  but  one  compensator,  as  above,  the  two  effects  must  be  small,  since  the  rays 
reflected  from  each  of  the  opaque  mirrors,  M  and  AT,  of  the  interferometer, 
must  eventually  enter  the  telescope,  to  unite  in  two  nearly  identical  images 
of  the  slit.  It  was  rather  unexpected  to  observe  that  the  interferences  are 
still  obtained,  even  when  the  two  slit  images  are  quite  appreciably  different 
in  size,  but  they  are  then  confined  to  a  single  plane,  as  will  be  shown  in  §  69. 


116 


THE   INTERFEROMETRY   OF 


Since  the  beam  of  light  coining  out  of  the  collimator  and  traversing  the 
grating  is  a  vertical  ribbon  of  light,  several  centimeters  high  vertically,  but 
very  thin  in  comparison  (a  few  millimeters)  horizontally,  it  is  relative  to  the 
vertical  plane  that  the  marked  effect  must  be  expected.  In  figure  81,  G  is 
the  grating,  cc  the  principal  plane  of  the  concave,  CD  that  of  the  convex  lens, 
M  the  opaque  mirror.  If  the  beam  consists  merely  of  the  axial  pencil  c,  the 
distorting  effect  due  to  the  introduction  of  the  lens  doublet  is  slight  for  any 
value  of  their  distance  apart,  D.  The  two  lenses  are  practically  equivalent 
to  a  plate.  If  a  broad  beam  dd  is  in  question  and  the  rays  retrace  their  path, 
the  same  is  still  true.  But  if,  on  changing  D,  the  rays  do  not  retrace  their 
path,  so  that  the  equivalent  lens  is  convergent  or  divergent,  then  the  rays 
after  leaving  M  re-impinge  on  the  grating  at  different  angles  than  before  and 
the  interference  pattern  is  correspondingly  changed,  principally  in  its  vertical 
relations. 

Thus  it  is  the  lens  system  which  changes  the  obliquity  of  rays  lying  in  a 
vertical  plane  and  passing  through  the  grating,  to  the  effect  that  the  axial  rays 
may  represent  a  case  of  either  maximum  or  minimum  path-difference.  The 
latter  will  be  the  case  when  the  divergent  pencil  which  usually  traverses  the 
grating  becomes  convergent  in  consequence  of  a  sufficiently  large  value  of  the 
D  of  the  lens  system. 


81 


83 


Jib  CV 


66.  Observations  largely  with  weak  lenses  and  short  interferometer.— The 

film  grating  used  (Wallace,  14,500  lines  to  the  inch)  was  cemented  with  Canada 
balsam  to  a  thick  piece  of  plate  glass,  so  that  the  total  thickness  of  plate  at  the 
grating  was  1,734  cm.  This  introduces  a  large  excess  of  path  in  one  of  the 
component  beams;  but  it  is  generally  necessary,  if  the  stationary  interferences, 
due  to  the  reflection  at  the  two  faces  of  the  plate  of  the  grating,  are  to  be  obvi- 
ated and  if  the  ellipses  produced  are  to  be  reasonably  large  for  adjustment  (cf . 
§  69).  The  lens  doublet  was  to  be  added  on  the  same  side  as  the  glass  speci- 
fied, so  that  the  excess  of  glass  thickness  on  one  side  was  further  increased  by 
about  0.19  cm.,  on  the  average.  Under  these  circumstances  the  ellipses  were 
strong,  but  (in  view  of  the  large  dispersion)  with  inconveniently  long  horizon- 

On  inserting  the  doublet  (convex  and  concave  lens,  each  i  meter  in  focal 

istance)  with  its  concave  lens  at  the  mirror  and  gradually  increasing  the 

distance  D  by  moving  the  convex  lens  toward  the  grating,  a  series  of  forms 


REVERSED   AND   NON-REVERSED    SPECTRA.  117 

was  obtained  which  passed  from  the  initial  horizontally  long  ellipse,  through 
circles,  vertically  long  ellipses,  vertical  lines,  into  hyperbolic  forms  of  increas- 
ing eccentricity,  as  recorded  in  figure  82. 

On  reversing  the  system,  keeping  the  convex  lens  fixed  near  the  mirror  and 
increasing  the  distance  D  by  moving  the  other  lens  toward  the  grating,  the 
original  ellipse  usually  flattened  out  further,  as  shown  in  figure  83.  Moving 
the  lenses  sideways  parallel  to  themselves  had  no  definite  effect;  moving 
them  fore  and  aft  together  (D  constant)  produced  results  similar  to  the  above. 
The  vertical  lines  of  figure  82  are  liable  to  be  sinuous  or  to  resemble  the  grain 
of  wood  around  a  knot.  In  case  of  figure  82,  as  the  equivalent  lens  lies  in 
front  of  the  mirror,  the  rays  reaching  the  grating  are  thus  necessarily  converg- 
ing. In  figure  83  the  equivalent  lens  lies  behind  the  mirror,  so  that  the  rays 
at  the  grating  are  more  convergent.  Both  positions  furnish  essentially 
convergent  rays. 

If  corresponding  to  figure  82,  the  convex  lens  is  kept  fixed  near  the  grating 
and  the  concave  lens  gradually  moved  up  to  it,  the  order  of  forms  is  reversed, 
but  not  quite  completely.  They  usually  terminate  in  long,  vertical  ellipses, 
before  reaching  which  the  wood-grained  forms  are  sometimes  passed.  The 
same  is  similarly  true  for  the  case  of  figure  83. 

With  cylindrical  lenses  (respectively  convex  and  concave,  each  i  meter  in 
focal  distance)  very  little  effect  was  observed  when  the  axes  of  the  cylinders 
were  parallel  to  the  slit.  With  the  axes  perpendicular  to  the  slit,  the  effects  of 
spherical  lenses  were  virtually  reproduced,  except  that  the  central  fields 
partook  of  a  more  rectangular  character. 

To  carry  out  the  purposes  of  the  present  paper  with  strong  lenses,  respec- 
tively convex  and  concave,  the  vertical  sheet  of  light  from  the  slit  must  be 
diverged  into  a  wedge  by  the  concave  lens  and  then  collimated  by  the  convex 
lens.  The  mirror,  normal  to  the  rays,  reflects  them,  so  that  they  retrace  their 
path  and  become  a  sheet  of  light  before  the  final  reflection  and  diffraction  at 
the  grating.  The  following  experiments  were  made  with  strong  lenses: 

At  first  lenses  of  double  the  preceding  focal  power,  /=  ±  50  cm.,  were  tried, 
but  with  no  essential  difference  in  the  results.  Thereupon  strong  lenses 
of  focal  distances /!=— 73  cm.  and /2=  13.1  cm.  were  used  together,  the 
convex  lens  being,  as  usual,  near  the  mirror.  For  .0  =  7.5  CEa-i  about,  these 
gave  fairly  clear  images  of  the  slit  and  it  was  easy  to  find  the  ellipses,  which 
were  now  very  eccentric,  almost  spindle-shaped  in  form.  They  could  be 
obtained  strong  and  clear  without  difficulty,  and  the  nearly  horizontal  lines 
filled  the  whole  spectrum.  Reversal  of  lenses  practically  failed  to  give  results, 
the  rays  after  reflection  being  too  divergent. 

On  the  large  interferometer,  where  the  distances  between  mirror  and  grating 
are  nearly  2  meters,  adjustment  was  more  difficult  and  the  result  (if  parallel 
rays  are  retained)  less  satisfactory,  because  the  slit  images  are  not  in  focus  at 
the  same  time.  This  is  particularly  the  case  when  the  convex  lens  is  nearest 
the  mirror  and  the  concave  lens  toward  the  grating.  Thus  when/=  ±  100  cm. 
and  D  =  1 5  cm.,  the  modified  slit  image  may  be  twice  as  large  as  the  other  and 


118  THE   INTERFEROMETRY   OF 

the  interferences  in  the  principal  focal  plane  of  the  telescope  are  only  just  seen. 
At  D  =  5  cm.,  however,  the  results  are  acceptable.  When  the  concave  lens  is 
nearest  to  the  mirror  and  the  convex  lens  toward  the  grating,  the  modified  slit 
image  is  smaller  than  the  other.  Adjustment  is  then  easier  and  the  usual 
elliptic  and  hyperbolic  forms  may  be  observed  without  trouble.  In  both  cases 
the  flickering  of  the  arc  lamp  used  passes  the  rays  through  different  parts  of 
the  lenses  relatively  to  the  center,  and  the  adjustment  is  thus  easily  destroyed. 
If  the  spectra  from  M  and  N,  however,  are  observed,  not  in  the  principal 
focal  plane  but  in  advance  of  it  (toward  the  eye),  interferences  of  great  interest 
will  be  observed,  to  be  discussed  in  §  69. 

67.  Remarks. — A  few  explanatory  observations  may  here  be  inserted.  The 
occurrence  of  the  elliptic  or  oval  and  the  hyperbolic  type  of  fringes  may  be 
most  easily  exhibited  by  laying  off  the  order  of  the  fringe  in  terms  of  the  dis- 
tance (in  arbitrary  units)  above  and  below  the  center  of  the  image  of  the  slit. 
If  we  call  the  latter  y  and  consider  the  allied  colors  of  thin  plates,  for  instance, 

n  =  2efi  cos  r/X  or  more  generally  n  =  (en/\)f  (y,  r) 

(where  e  is  the  thickness  of  the  plate,  p.  its  index  of  refraction,  X  the  wave- 
length of  light  in  case  of  a  dark  locus  of  the  order  n)  is  to  be  expressed  in  terms 
of  y,  which  itself  determines  e  cos  r,  r  being  the  angle  of  refraction  in  the  plate 
of  the  grating.  The  phenomenon  will  thus  be  coarser  for  red  light  than  for 
violet  light,  since  n  decreases  when  X  increases,  and  any  two  curves,  r  and  v, 
figure  84,  may  be  assumed  as  the  loci  of  the  equation  in  question.  If,  now, 
horizontal  lines  be  drawn  for  n=  i,  2,  3,  etc.,  they  will  determine  the  number 
of  dark  bands  in  the  spectrum  for  any  value  of  y. 

If  the  central  ray  is  also  a  line  of  symmetry  and  intersects  the  grating  nor- 
mally, it  must  correspond  to  a  maximum  or  a  minimum  of  n.  These  conditions 
are  shown  in  the  diagram  at  M,  where  the  maximum  number  of  bands  occurs, 
and  at  m,  where  the  reverse  is  true.  The  question  is  thus  referred  to  two  sets 
of  loci,  rr'  and  vo't  or  r'r"  and  v'v*,  etc.  In  the  former  case  e  cos  r  varies  with  y 
in  the  same  sense  as  /z/X;  in  the  latter  in  the  opposite  sense  and  is  preponder- 
ating in  amount.  Both  may  vary  at  the  same  rates  in  the  transitional  case,  in 
which,  therefore,  the  two  curves  r  and  v  are  at  the  same  distance  apart  for  all 
values  of  y. 

Suppose,  furthermore,  that  the  same  phenomenon  is  exhibited  in  terms  of 
wave-length  X,  as  in  the  lower  part  of  the  diagram,  the  spectrum  being  now 
equally  wide  for  all  values  of  y,  while  at  any  given  y  the  upper  diagram  still 
shows  the  number  of  dark  points  (bands)  between  r  and  v.  If  now,  we  suppose 
that  under  any  conditions  these  dark  points  are  grouped  symmetrically  with 
reference  to  any  given  color  (which  is  probable,  for  a  maximum  or  a  minimum 
of  any  value  of  y  will  be  so  for  all  values),  and  that  the  successive  dark  points 
have  been  connected  by  a  curve,  the  interference  pattern  will  be  of  the  elliptic 
type  in  case  of  aa't  a"a"f,  and  of  the  hyperbolic  in  the  case  of  a' a". 

The  other  features  of  the  phenomenon  are  secondary  and  therefore  left  out 


REVERSED   AND   NON-REVERSED   SPECTRA. 


119 


of  the  diagram.  Thus,  for  instance,  the  distance  apart  of  the  bands  shrinks 
from  red  to  violet,  and  the  ovals,  etc.,  are  only  appreciably  symmetric,  because 
they  occupy  so  small  a  part  of  the  spectrum.  The  horizontal  distribution  of 
dark  bands  around  the  center  is  determined  by  variations  e  cos  r  and  is  not 
linear.  Whether  the  long  axes  of  the  ellipses  are  horizontal  or  vertical  depends 


upon  the  slope  of  the  lines  r  and  v.  Maxima  and  minima  will  not,  as  a  rule, 
occur  close  together,  though  in  certain  wood-grain-shaped  patterns  this  seems 
to  be  the  case. 

In  conclusion,  therefore,  the  main  feature  in  modifying  the  type  of  inter- 
ference pattern  is  the  varying  thickness  of  the  compensator.  For  oval  types 
the  preponderating  lens  is  convex;  for  the  hyperbolic  type  it  is  concave. 
Neither  of  these  lenses  is  here  appreciably  affected  in  modifying  the  horizontal 
distribution  of  path-difference,  because  the  dispersion  of  the  grating  requires 
a  horizontally  parallel  system  of  rays. 

68.  Observation  with  lens  systems  on  both  sides. — The  method  shown  in 
plan  in  figure  85  (L  and  U  convex  lenses,  G  grating,  M  and  N  mirrors,  telescope 
at  T)  was  tested.  The  outcome  can  not  at  once  be  foreseen,  since  the  focal 
distances  for  different  colors  is  different  and  since  slight  displacements  of 
either  lens  must  greatly  modify  the  interference  pattern.  The  latter,  however, 
as  obtained  in  every  case,  proved  to  be  exceedingly  fine  lines,  tipping  in  the 


120 


THE   INTERFEROMETRY  OF 


usual  way  with  the  motion  of  the  micrometer  and  indicating  a  center  of  ellipses 
very  distant  in  the  field  of  the  spectrum.  In  other  words,  the  interference 
pattern  is  no  longer  automatically  centered  and  is  therefore  useless. 

A  modification  of  this  plan  is  the  method  shown  in  figure  86  (horizontal 
section),  where  B  is  the  beam  from  the  collimator,  L,  L',L",L"',  four  condens- 
ing lenses  of  the  same  power  (/=  50  cm.),  G  the  grating,  Mand  N  opaque  plane 
mirrors,  T  the  telescope.  In  all  the  above  cases  the  horizontal  rays  from  the 
collimator  traverse  the  grating  in  parallel  and  eventually  condense  to  a  single 
point  in  the  field  of  the  telescope.  The  same  is  true  of  all  rays  having  the  same 
angle  of  altitude.  These  rays,  therefore,  act  as  a  whole,  since  they  pass  through 
the  plate  of  the  grating  at  the  same  angle  of  incidence.  On  the  other  hand, 


relative  to  a  vertical  plane,  the  rays  traverse  the  grating  at  different  angles, 
each  angle  corresponding  to  a  horizontal  strip  of  the  spectrum.  It  is  by  the 
easy  modification  of  this  obliquity  that  the  curved  compensator  becomes  effec- 
tive. In  figure  86  the  rays  are  also  oblique  relative  to  a  horizontal  plane ;  but 
the  result,  unfortunately,  is  not  available,  since  each  of  these  oblique  rays  must 
have  its  own  complete  spectrum.  Consequently  the  diffracted  pencil  will  con- 
sist of  an  infinite  number  of  overlapping  spectra,  the  extreme  cases  lying  within 
the  same  angle  a  shown  in  the  figure.  A  large  telescopic  objective  would  then 
reunite  these  spectra  into  a  white  image  of  the  slit,  while  a  small  objective  will 
show  colored  slit  images,  passing  from  impure  red  to  impure  violet.  Naturally, 
the  interferences  will  also  overlap,  and  therefore  vanish. 

69.  Telescopic  interferences.— If  interference  patterns  of  small  angular 
extent  are  to  be  obtained,  it  is  essential  that  the  rate  at  which  obliquity 
increases  from  ray  to  ray  be  made  as  large  as  practicable.  Probably,  therefore, 


REVERSED  AND   NON-REVERSED   SPECTRA.  121 

an  opportunity  for  realizing  these  conditions  will  be  found  within  the  telescope ; 
i.e.,  after  the  rays  pass  the  objective.  The  endeavor  would  therefore  be 
directed  to  bringing  two  spectra,  focussed  in  two  planes,  one  of  which  is  behind 
the  other  and  consequently  of  different  sizes,  both  vertically  and  horizontally, 
to  eventual  interference. 

The  experiment  was  made  on  the  long  interferometer  (fig.  87),  the  distances 
between  mirror  M  and  grating  G  and  from  the  latter  to  the  mirror  N  being 
nearly  2  meters  each.  C  is  the  lenticular  compensator,  consisting  of  two  lenses, 
respectively  concave  and  convex,  each  having  the  same  focal  distance,  /=  =±=  50 
cm.  The  distances  apart,  D,  of  the  lenses  may  be  varied.  The  glass  plate  C', 
which  is  revolvable  about  the  vertical,  is  thick  enough  to  exactly  counterbal- 
ance, if  necessary,  the  thickness  of  the  glass  plate  of  the  grating  and  of  the  lens 
system  C.  A  sharp  wedge  sliding  transversely  may  also  be  used.  It  is  best  to 
replace  C'  by  two  plates  of  glass,  one  thick  and  the  other  thin,  so  that  the  lat- 
ter may  be  removed. 

The  telescope  directed  along  the  axis  R  will  therefore,  in  general,  see  two 
white  slit  images,  A  and  A'  (fig.88),  not  both  in  focus  at  once,  A'  coming  from 

cA-     $0? 

(,' 
Jc' 
88 

M  being  larger,  A  from  N  (parallel  rays)  smaller.  The  focal  plane  of  A'  will  be 
towards  the  grating  as  compared  with  A,  and  A'  is  larger  than  A,  in  proportion 
as  the  distance  apart  of  the  lenses  C  is  larger.  Similarly,  the  two  spectra 
are  observed  along  the  diffraction  axis,  D,  not  in  focus  at  once  and  of 
different  areas. 

To  obtain  the  interferences  the  slit  image  A  must  be  placed  anywhere  within 
A',  and  they  will  occur  at  the  top  of  the  spectrum  if  a  and  a'  are  vertically  in 
coincidence ;  in  the  middle  if  b  and  b'  coincide,  etc. 

The  plane  of  the  new  interferences  is  no  longer  the  principal  focal  plane,  con- 
taining the  Fraunhofer  lines,  but  lies  in  front  of  it;  i.e.,  towards  the  eye  of  the 
observer  and  away  from  the  grating.  This  distance,  measured  along  D  for  the 
given  small  telescope  used,  was  fully  i  cm.  The  focal  planes  of  the  two  spectra 
are  usually  not  so  far  apart.  A'  corresponds  to  a  virtual  object  behind  the 
observer. 

If  the  vertical  plane  in  which  the  interferences  lie  be  taken  as  the  image,  the 
object  would  be  situated  about  3  meters  beyond  the  objective  of  the  telescope 
used.  This  would  place  it  30  cm.  in  front  of  the  mirror  M  or  N,  where  there  is 
but  a  single  beam  in  each  case.  In  fact,  the  telescope  may  be  brought  quite 


122  THE   INTERFEROMETRY  OF 

up  to  the  grating.  Hence  interference  is  produced  in  the  telescope  itself,  where 
rays  are  relatively  very  divergent,  a  condition  which  accounts  for  the  smallness 
of  the  interference  pattern.  This  understanding  of  the  case  is  tentatively 
shown  in  figure  89,  where  0  is  the  objective  of  the  telescope,  M  the  larger  image 
from  the  mirror  with  the  lens  compensator,  and  AT  the  image  from  the  other 
mirror  (parallel  rays) .  If  the  corresponding  rays  be  drawn  through  the  extrem- 
ity of  M  and  N,  their  fields  of  interference,  F  and  Fr,  would  begin  in  the  plane 
I  and  /'.  For  axial  rays  it  would  be  at  i.  Thus  the  locus  as  a  whole  would  not 
be  a  plane,  and  this  seems  to  be  the  case.  If  the  telescope  moves  toward  the 
grating,  II'  moves  toward  the  right  in  the  figure,  as  though  the  virtual  object 
beyond  the  grating  were  fixed  in  position.  At  all  events,  the  problem  is  to  find 
the  interference  diagram  of  two  symmetrical  plane  parallel  spectra,  of  different 
areas  and  placed  at  definite  distances  apart. 


The  appearance  of  the  fringes  is  indicated  in  figure  90,  where  5  is  the  height 
of  the  spectrum,  usually  quite  out  of  focus.  There  are  many  more  lines  than 
could  be  drawn  in  the  sketch.  The  ends  a  and  a'  seem  to  surround  small  ellipses, 
but  these  are  not  quite  closed  on  the  outer  edge.  The  center  of  symmetry 
is  at  C.  The  demarcations  are  stronger  and  broader  vertically  if  the  distance 
apart  of  the  lenses  C  (fig.  87)  is  small;  fainter,  but  nevertheless  clear  and  nar- 
rower, if  this  distance  is  large.  Horizontally  the  fine  lines  thread  the  spectrum. 
The  best  results  were  obtained  when  the  lenses  C  are  less  than  i  cm.  apart,  the 
middle  band  being  about  half  as  high  as  the  spectrum.  Two  contiguous  lenses 
gave  a  design  which  nearly  filled  the  spectrum  vertically.  For  practical  pur- 
poses the  lens  compensator  C  is  to  be  attached  to  the  mirror  M,  just  in  front 
of  and  moving  with  it .  It  makes  little  difference  here  whether  the  concave  lens 
or  the  convex  lens  of  the  doublet  C  is  foremost. 

If  the  micrometer  M  is  moved,  or  if  the  telescope  is  slid  to  the  right  or  left, 
or  forward,  so  as  to  take  in  other  parts  of  the  spectrum,  the  nearly  closed  lines 
at  a  and  a'  become  finer  and  finer  crescent-shaped  lines, 
always  open  outward,  till  they  pass  beyond  the  range  of 
vision.  The  whole  phenomenon  remains  on  the  same  level 
of  the  spectrum.  On  moving  the  telescope  forward  as  far 
as  G  (fig.  87),  the  ocular  has  to  be  drawn  outward  (towards 
the  eye)  till  it  is  fully  2  cm.  beyond  the  position  of  the 
principal  focal  plane.  The  whole  spectrum  is  now  seen 
with  the  interferences  from  red  to  violet  (no  ellipses),  but  Q 
having  the  same  relative  position  as  before.  The  central  "  ' 
horizontal  band  measures  about  one-fifth  the  height  of  the  spectrum,  while 
the  fine  parallel  horizontal  lines  extend  to  the  upper  and  lower  edges.  The 


REVERSED  AND   NON-REVERSED   SPECTRA.  123 

appearance  is  now  curiously  like  a  blunt  wedge  (fig.  91),  with  a  band  at  b 
nearest  the  eye,  and  the  lines  dd  extending  quite  to  the  rear.  This  impression 
is  probably  an  illusion,  due  to  the  shading;  the  lines  grow  finer  and  are  more 
crowded  toward  the  bottom  and  top  of  the  spectrum.  The  illusion  of  a 
reentrant  wedge  is  not  possible. 

To  use  this  interference  pattern  for  measurement,  the  cross-hair  is  supposed 
to  pass  through  the  region  c  (fig.  90)  symmetrically.  Very  slight  motion  of  the 
micrometer  mirror  M  then  throws  c  either  to  the  right  or  the  left  of  the  cross- 
hair. In  this  case  the  lens  doublet,  C,  is  attached  to  the  mirror  and  moves  with 
it,  as  stated.  To  obtain  the  extreme  of  sensitiveness,  the  path-difference  of  NG 
and  GM  must  be  all  but  zero;  i.e.,  the  grating  plate  G  and  the  lens  doublet  C 
(fig.  87)  must  be  all  but  compensated  for  equal  air-distances  by  the  compen- 
sator C.  In  this  case  of  full  compensation,  the  interference  pattern,  in  the 
absence  o£  a  doublet  C,  would  be  enormous  and  diffuse,  seen  preferably  in  the 
principal  plane  of  the  telescope,  but  useless  for  measurement.  The  introduc- 
tion of  a  lenticular  compensator,  balanced  by  a  compensator  in  GN,  transforms 
the  huge  pattern  into  the  small  interference  fringes  in  question,  with  the  advan- 
tage that  the  high  mobility  of  the  coarse  design  has  been  retained.  In  other 
words,  an  index  suitable  for  adjustment  has  been  found,  compatible  with 
extreme  sensitiveness.  In  fact,  it  is  difficult  to  place  the  micrometer  mirror 
M  so  that  the  region  c  (fig.  90)  is  exactly  bisected.  As  the  plane  in  which  these 
interferences  are  seen  most  distinctly  is  i  cm.  or  more  anterior  to  the  principal 
focal  plane,  the  Fraunhofer  lines  are  unfortunately  blurred  and  a  cross-hair  is 
needed  as  a  line  of  reference. 

I  may  in  conclusion  refer  to  a  similar  series  of  experiments  now  in  prog- 
ress, in  which  the  compensators  placed  in  the  M  and  N  pencils  (fig.  87, 
C,  C'),  instead  of  being  of  different  shapes  as  above,  are  plates  of  different 
kinds  of  glass  (crown  and  flint,  for  instance).  Here  the  successive  differ- 
ences of  dispersive  power,  from  wave-length  to  wave-length,  produce  effects 
closely  resembling  those  discussed,  with  the  advantage  that  difficulties 
inherent  in  the  curved  system  are  avoided. 


CHAPTER  X. 


THE  DISPERSION  OF  AIR. 

70.  Introduction. — In  view  of  the  long-armed  interferometer  available,  it 
seemed  interesting  to  test  the  refraction  of  air  at  different  wave-lengths,  X.  An 
iron  tube  of  inch  gas-pipe,  138  cm.  long,  was  therefore  placed  in  one  or  the  other 
of  the  component  beams.  The  tube  was  closed  at  both  ends  by  glass  plates, 
about  one-eighth  of  an  inch  thick,  kept  in  place  with  resinous  cement.  A 
lateral  tube  communicated  with  an  air-pump  and  drying  train,  so  that  the 
tube  could  be  alternately  exhausted  and  refilled  with  air.  By  using  sun- 
light, the  different  lines  of  the  spectrum  were  obtained  with  sufficient  clear- 
ness, and  the  method  consisted  in  finding  the  reading  of  the  micrometer  for 
successive  Fraunhofer  lines,  both  for  the  case  of  a  plenum  of  air  and  for  a 
vacuum.  If  AN  is  the  (monochromatic)  displacement  of  micrometer  corre- 
sponding to  the  latter  difference  of  pressure,  M  being  the  index  of  refraction 
of  air,  e  the  thickness, 

(i)  AAf> 

To  determine  MX,  we  must  know  d^/  d\.  It  has  been  omitted  above,  because 
it  enters  differentially  and  because  of  its  small  value.  It  appears  as  a  con- 
stant decrement  of  AA/x,  as  X  is  constant  and  dMx/  d\  is  negative.  In  the  present 
case,  where  M  is  actually  to  be  measured,  dnj d\  enters  directly  and  is  essen- 
tial; but  it  follows  from  any  two  experiments  when  M  is  found  for  different 
colors. 

TABLE  8. — Values  of  B.    Inch  iron  gas-pipe,  138.0  cm.  long.    D  line. 


t 

Bar 

P 

1&&N 

22.0° 

76.20\ 

75-o 

38.0 

20°     / 

22.3° 

75.831 

74-5 

37-9 

37-7 

20°     / 

37-6 

37-9 

37-7 

37-6 

37-6 

37-7 

37-5 

37-7 

Mean.  .  .  . 

37.69 

71.  Observations  with  arc  lamp — ln  table  8  results  are  given  as  obtained 
with  the  electric  arc,  in  which  the  sodium  line  usually  appears  with  sufficient 
distinctness  in  the  spectrum  to  be  available  as  a  line  of  reference  for  measure- 
124 


REVERSED   AND   NON-REVERSED   SPECTRA.  125 

ment.  Disregarding  earlier  results,  the  following  are  mean  values  of  the  ten 
independent  data  for  A/V  (each  comprising  a  reading  for  vacuum  and  for 
plenum)  : 


t  =  22.  3       £  =  74.5011. 
Thus 


where  ^  refers  to  normal  pressure  and  absolute  temperature  (r).  If  /KO  is 
given  for  the  D  line,  d/i/  aX  is  determinable.  It  will  be  sufficient  for  the  present 
purposes  to  put  t^—A-{-B/\z,  or  X.aju/aX  =  —  zB/X2 


5  referring  to  r  and  p.     Mascart's  *  value  for  ^—  i  (agreeing  with  Fabry's) 
is  icreX292.7,  whence 

£=icr14Xi.34  at  r  and  p 

If  the  value  B  be  computed  from  Mascart's  observations  between  C  and 
£,  D  and  F,  respectively, 


so  that  the  mean  value  io14S=i.6s  may  be  taken.    Since  the  last  decimals 
of  /z  are  in  question,  it  will  not  be  correct  to  more  than  5  to  10  per  cent. 

The  value  found  above  (io14B  =  i.34)  is  therefore  somewhat  too  small. 
True,  since  from  equation  (3) 

(4) 


an  error  of  icr4  cm.  in  AN  is  an  error  of  0.13  X  io~14  or  10  per  cent  in  B,  Very 
close  agreement  can  not  therefore  be  expected  in  either  result.  One  is  tempted 
to  refer  the  present  low  value  of  B  to  flexure  of  the  glass  end  plates  of  the 
tube,  which,  when  the  tube  is  exhausted,  become  slightly  saucer-shaped  and 
introduce  a  sharp  concentric  wedge  of  glass  into  the  component  beam,  whereby 
the  interference  pattern  is  changed,  probably  in  the  direction  of  smaller 
values,  as  found.  But  the  direct  experiments  below  do  not  show  this.  In 
any  case,  the  measurement  of  B  lies  at  the  limits  of  the  method.  An  advan- 
tage may  possibly  be  secured  by  using  two  identical  tubes,  one  in  each  com- 
ponent beam,  the  tubes  to  be  exhausted  alternately.  The  sensitiveness 
would  then  be  doubled. 

72.  Observations  with  sunlight.  Single  tube.  —  These  observations  are 
given  in  table  9,  the  exhaustion  throughout  being  75  cm.  and  the  temperature 
about  1  6°.  In  the  first  set  sunlight  was  used  without  a  condensing  lens;  in 

*  See  excellent  summary  in  Landolt  and  Boernstein's  Tables,  1905,  p.  214. 


126 


THE   INTERFEROMETRY   OF 


the  second  set  the  sun  was  focussed  with  a  weak  lens  (0.5  meter  in  focus)  at 
the  point  formerly  occupied  by  the  electric  arc.  The  spectrum  (particularly 
in  the  second  case)  was  brilliant  and  the  lines  clear.  The  focus  of  sunlight  is  to 
be  placed  just  outside  the  focus  of  the  collimator  lens,  in  order  that  a  nearly 
linear  pencil  may  be  available  to  penetrate  the  long  refraction  tube  twice. 
The  distance  of  the  collimator  lens  to  the  grating  was  about  2  meters.  The 
spectrum  is  then  a  bright  band  in  the  telescope,  the  width  being  limited  by  the 
height  of  the  ruled  part  of  the  grating.  The  strip  of  white  light  on  the  grating 
should  not  be  more  than  a  few  millimeters  wide.  It  must  therefore  be  nar- 
rowed by  an  opaque  screen  (wide  slit  of  the  given  width)  in  the  path  of  the 
beam  (see  fig.  92  below). 
TABLE  9. — Dispersion  of  air.  Tube /=  138.0  cm.  Bar.  77.25  cm.  at  19.5°.  £=75.0  cm 


Line. 

Temp. 

io*AN 

io+145 

C 
D 
b 

16.0° 

38.5] 
39- 
394J 

i-5 

C 
D 
b 

1  6.0° 

38.5] 
38.9 
39-3J 

1.4 

Improved  seeing,  weak  condensing  lens. 

C 
D 

16.4° 

38.81 
39-2J 

I-5I 

C 
D 
b 
F 

16.4° 

38.9] 
39-1 
?39-3 
40.  1  J 

1.65 
1.4 

C 
F 

16.4° 

38.81 
40.  1  / 

1.61 

The  equation  for  B  in  this  case,  if  the  symbol  8  refers  to  differences  for  two 
given  values  of  X,  is 


P   273 


8L 
V 


if  the  value  of  B  is  to  hold  for  normal  conditions. 

The  data  are  shown  in  table  9,  series  i  and  2  being  obtained  without  con- 
densing lens.  These  are  inferior,  as  regards  definition  of  lines,  to  the  subse- 
quent set,  in  which  condensed  sunlight  was  used.  In  all  cases  there  is  some- 
times an  irregularity  (marked  by  ?  in  table  9)  in  which  the  observation  is 
obviously  discordant,  but  the  reason  could  not  be  found.  Possibly  values  of 
r  and  p  taken  were  not  the  actual  values.  The  data  for  B  X  io14  given  in  the 
table  are  mean  values.  Some  of  these  are  low.  Later  values,  where  the  F 
line  is  included,  come  out  larger,  the  range  being  from  1.3  to  1.8  or  1.5,  on 
average.  It  is  desirable  to  use  the  whole  of  the  available  range  of  the 
spectrum  (sufficiently  luminous  from  C  to  F)  to  obtain  an  acceptable  value 
of  the  coefficient  B  and  additionally  to  improve  the  method  by  using  two 


REVERSED   AND   NON-REVERSED   SPECTRA.  127 

identical  tubes,  alternately  exhausted  as  suggested  above.  The  attempted  B 
measurement  is  at  the  limits  of  the  method,  as  has  already  been  instanced 
in  the  discussion  of  errors  in  the  preceding  paragraph,  and  it  is  not  to  be  con- 
cluded that  data  which  happen  to  agree  with  Mascart's  result  from  a  correct 
application  of  the  present  method.  In  fact,  there  is  no  reason  for  excluding  the 
exceptional  values,  and  the  present  results  are  to  be  regarded  as  preliminary. 

73.  Two  (differential)  refraction  tubes. — In  the  following  experiments  two 
identical  iron  tubes  (138  cm.  long,  of  inch  gas-pipe)  were  installed,  one  being 
placed  in  each  of  the  component  beams  of  light,  which  subsequently  interfered, 
and  the  tubes  were  exhausted  alternately.  There  are  apparently  three  advan- 
tages in  this  arrangement.  In  the  first  place,  the  sensitiveness  is  doubled;  in 
the  second,  the  flexure  of  glass  plate  should  be  the  same  at  each  tube,  in  each 
experiment,  and  thus  fail  to  disturb  the  interference  pattern.  Furthermore, 
by  using  the  tubes  in  parallel  (*.*.,  exhausting  both  at  the  same  time),  any 
irregularity  of  flexure  effect,  etc.,  should  be  determinable,  as  the  air  in  both 
tubes  will  be  identically  circumstanced.  Finally,  the  air  being  inclosed  in  a 
thick  metallic  envelope  at  both  beams  is  not  subject  to  incidental  disturb- 
ances. An  unexpected  difficulty,  however,  was  encountered ;  for  there  is  reflec- 
tion of  direct  spectra  from  the  eight  glass  surfaces,  and  this  must  be  specially 
met.  The  direct  spectrum  is  easily  eliminated  by  inclining  the  grating  until 
the  reflected  interference  spectra  are  at  a  different  level;  but  reflections  of 
this  spectrum  are  not  so  easily  dealt  with.  Fortunately  they  are  weak.  Even 
so,  they  are  very  annoying,  as  they  overlap  the  interference  pattern  and  dull 
it.  They  could  be  eliminated  by  attaching  the  glass  plates  obliquely  to  the 
axis  of  the  pipes,  but  this  remedy  was  not  thought  of  at  the  outset. 


Figure  92  is  a  diagram  of  the  disposition  of  the  parts  of  the  apparatus. 
L  is  the  beam  of  white  sunlight  from  the  collimator  limited  laterally  by  the 
wide  slit  (1  inch)  5.  G  is  the  grating,  T  and  T'  the  two  refraction  tubes, 
M  (micrometer)  and  N  the  opaque  mirrors,  R  the  refracted  and  D  the  dif- 
fracted (spectrum)  beam  of  light.  C  is  virtually  a  four-way  stopcock  (or  two 
3 -way  glass  stopcocks)  leading  respectively  to  the  exhaust  pump  E  and  dry 


128 


THE    INTERFEROMETRY   OF 


air  supply  A,  from  the  tubulures  e  and  e'  of  both  refraction  tubes  T  and  T. 
These  are  therefore  alternately  exhausted. 

Preliminary  results  are  given  in  table  10,  the  arc  lamp  with  its  sodium  line 
being  used  in  the  absence  of  sunlight.    It  will  be  seen  that  AN,  apart  from 
temperature  (which  is  here  higher  than  above),  has  been  doubled.     The 
deflections  were  symmetrical  within  o.isX  icr3  cm. 
TABLE  10. — Dispersion  of  air.    Differential  tubes,  each  138  cm.  long.    D  line  in  electric  arc. 


Barometer. 

Temp. 

lo'Atf 

(*>-l)Xio« 

-BoXio14 

75.93  cm.,  22° 
76.02  cm.,  19.5° 

21-5° 

18.0° 

754 
75-3 

g 

291.6 
291.8 

1-47 
1.49 

(Single  lube) 

23.0° 

37^8 

293-6 

i.  80 

The  values  of  B  found  in  the  first  two  series  of  this  table,  if  the  standard 
value  of  HQ—  i  =0.0002927  is  assumed,  is  somewhat  small,  but  as  near  to  the 
true  values  as  may  be  expected.  Again,  if  jE?Xio14=i.6s  is  assumed,  the 
Aio—  i  values  given  in  the  table  are  similarly  small,  being  0.3  per  cent  short  of 
standard  values.  A  single-tube  experiment  made  for  comparison  (series  3), 
similarly,  came  out  too  large  in  each  case.  It  follows  from  this  that  p  and  t 
observations  are  not  sufficiently  guaranteed.  It  is  hardly  probable,  however, 
that  with  a  micrometer  reading  to  iQ-3  cm.  and  estimated  to  lo"4  cm.  (vernier) 
the  precision  can  be  much  enhanced ;  for  since 
X'AAT  76  T 


dJ5o=_VAA[^-r_ 

dp         2     e    273  p* 

or  at  the  C  line  and  D  lines,  respectively, 

8p  =  i  cm.         8B  - 


dT 


2    e    273 p 


fo.2Xio-14 
1o.2Xio-14 


Now,  unless  the  measurement  can  be  made  in  terms  of  rings,  it  is  difficult 
to  detect  a  few  millimeters  of  pressure  difference  by  displacement  only. 

The  interesting  question  now  occurs  whether  the  two  tubes,  if  identical 
for  a  plenum  of  air,  remain  identical  (no  shift  of  the  interference  pattern) 
throughout  all  the  stages  of  identical  exhaustion.  On  trial,  nothing  could  be 
detected,  the  fringes  remaining  stationary  during  the  whole  period  of  exhaus- 
tion, or  during  the  influx  of  air  following  a  high  vacuum.  Hence  there  is  no 
perceptible  difference  effect  of  flexure  of  the  glass  ends,  and  the  ultimate 
question  of  accuracy  depends  on  the  measurement  of  r  and  p.  To  eliminate 
the  possible  effect  of  flexure,  an  air  column  of  negligible  length,  in  which  the 
glass  effect  only  is  present,  will  have  to  be  tested;  otherwise  there  is  no  possi- 
bility of  separating  the  air  and  glass  effect. 


REVERSED   AND   NON-REVERSED   SPECTRA. 


129 


74.  Differential  and  single  refraction  tubes.  Sunlight. — The  direct  experi- 
ments for  the  coefficient  B  were  now  resumed  and  conducted  with  sunlight, 
with  the  results  given  in  table  1 1 .  The  first  two  series  were  made  with  the 
two  identical  tubes  specified,  exhausted  alternately,  one  tube  containing  a 
plenum  of  air,  while  the  other  was  nearly  empty  in  each  experiment.  The  C 
and  F  lines  alone  were  used  for  measurement.  In  spite  of  the  large  displace- 
ment (AAT=o.o76  to  0.078  cm.),  the  results  were  not  as  satisfactory  as  was 
expected,  owing  to  the  fact  that  sharpness  of  vision  is  made  difficult  by  the 
stray  reflected  spectra  to  which  reference  has  already  been  made.  But  the 
data  for  B  obtained  with  one  exception  (No.  2  in  the  first  series)  are  consistent 
and  reasonably  good.  In  every  other  respect  the  work  was  satisfactory  and 
could  have  been  improved  by  using  oblique  cover-glasses.  The  values  of  B 
obtained  are  therefore  disconcerting. 

TABLE  ir. — Dispersion  of  air.  First  and  second  series,  Differential  Tubes,  each  138.0  cm. 
long.    C  and  F  lines.    Third  and  fourth  series,  Single  Tube,  good  adjustment. 


Barom.        p 

Line 

i&AN 

iouB 

Temp. 

Barom.        p 

Line 

lo'AiV 

iouB 

Temp. 

cm. 
77-05\      74-0 

22°      / 

74.0 
74.0 
74.0 

C 
F 
C 
F 
C 
F 
C 
F 

cm. 
76.4 
78.0 
76.1 
78.2 

76.3 
78.0 
76.2 
78  o 

I.O 

1-3 

i.i 
i.i 

°C 
17.4 

17.4 
17.4 
17.4 

cm. 
77.22}      74.0 

21°   / 

74.0 
74.0 
74.0 

C 
F 
C 
F 
C 
F 
C 
F 

cm. 
38.4 
39-2 
38.2 
39-3 
38.3 
39-3 
38.2 
39-O 

0.99 
i.  ii 
i.i  8 
i.  06 

°C 
18.0 

1  8.0 
1  8.0 
18.0 

76.27\     74.0 

23°    / 
74.0 

74.0 

C 
F 
C 
F 
C 
F 

75-8 
774 
75-4 
77.2 

75-4 
77  * 

I.O 

I.I 
I.I 

21.4 
21.4 
21.4 

77.22!      76.0 

21°   / 
76.0 

76.O 

C 
F 
C 
F 
C 
F 

39-3 
40.1 
39-2 
40.0 
39-0 
40.0 

0.99 
•99 
•99 

18.1 
18.1 
18.2 

76.0 

C 

F 

39-2 
40  o 

•99 

18.2 

I  then  went  back  to  the  single-tube  experiments  (in  series  3  and  4),  and 
these  are  the  smoothest  results  obtained.  The  C  and  F  lines  were  used  as 
before.  In  the  last  series,  for  instance,  the  micrometer  reading  is  the  same 
to  io~*  cm.  throughout.  In  spite  of  this  satisfactory  behavior,  the  value  of 
B  obtained  is  again  of  the  same  low  order,  all  the  data,  both  for  double  and 
single  tubes,  being  consistent  throughout  in  this  respect.  Series  3  and  4  agree, 
although  p  is  changed  from  74  cm.  to  76  cm.  of  mercury. 

In  table  12  I  have  summarized  the  data  in  comparison  with  the  standard 
results,  computing  /J.Q—  i  and  B0,  for  each  of  the  cases,  reducing  all  values  to 
o°C.  and  76  cm.  of  mercury.  The  difference  of  ju<>—  i  for  the  F  and  C  lines, 
which  is  3.2  X  icr6  for  the  standard  data,  is  but  2.3 X  icr6  on  the  average  in  the 
present  results.  Similarly,  the  mean  values  of  the  latter  are  io-*X4-i  and 
io~'X3.2  larger  for  the  C  and  F  lines,  respectively,  than  the  standard  results. 


130 


THE   INTERFEROMETRY   OF 


These  conditions  are  particularly  puzzling,  since  in  §73,  with  the  use  of  the 
arc  lamp,  both  results  were  nearly  normal.  I  therefore  endeavored  to  detect 
the  causes  for  this  difference  of  behavior. 

TABLE  12. — Summary  of  Table  n.    "St."  refers  to  standard  data.    A=  (MO— i). 


Line,  etc. 

XX  io6 

St. 
io'A 

Series. 

Ser. 
(i)to(4) 
-BoXio" 

(MO-  i)  St. 

(I) 
io«  A 

(2) 
IOM 

(3) 
loM 

(4) 

IOM 

SoXio14C. 

-BoX  io"P. 

c 

65-63 
48'.6i 

291.8 
295.0 

295.0 
-3-2 
297-3 
-3-3 

2.3 

296.0  i  296.3 
-4.2  I   -4.5 

298.0  ;  298.7 

-3-3  I  -3-7 

2.3      2.4 

296.5 
-4-7 
298.6 
-3-6 

2.1 

1.18 

1.20 
1.28 
1.  08 

1.85 
2.37 

2.22 

2-55 

1-33 
1.56 
1.70 
1.50 

10*  Diff  .  from  St. 
p 

io»Diff.  from  St. 
lo'Diff.  FandC. 

3-2 

The  standard  of  length  was  first  compared  with  a  normal  meter,  showing 
that  the  i  =  138.0  cm.  for  the  M  tube  should  be  replaced  by  137.75  cm.  and 
for  the  N  tube  by  137.59  cm-  As  this  correction  affects  all  the  results,  /*0—  i 
and  B,  in  the  same  ratio,  it  contributes  nothing  to  modify  the  discrepancy 
in  question.  The  correctness  of  the  micrometer  screw  was  assumed,  as  it  was 
of  good  manufacture. 

The  test  next  was  made  to  see  if  the  lines  taken  as  C  and  F  actually  had 
the  accepted  wave-lengths.  A  revolvable  arm,  with  its  axis  at  the  grating 
and  125.5  c111-  long,  was  therefore  installed  for  the  direct  measurement  of  the 
diffraction  of  the  grating.  The  results  obtained  for  the  wave-lengths  of  the 
lines  taken  showed  that  no  mistake  had  been  made  in  their  selection. 

To  endeavor  to  obtain  further  evidence,  the  values  of  B  were  computed  for 
the  mean  data  of  table  n,  by  using  the  standard  values  for  HQ—  i  in  case  of 
the  C  and  F  lines  and  the  AAT/e  given  by  the  observations.  The  results  so 
obtained  are  given  in  the  last  columns  of  table  12,  for  each  of  the  four  series 
and  the  C  and  F  lines  at  normal  temperature  and  pressure.  The  mean  results 
are  thus — 

#oXio14=i.i9  from  observations  with  sunlight  directly 

between  C  and  F  lines. 

B0X  io14=2.25  from  standard  ;u0—  i  =  .0002918  at  C  line. 
B0X  io14=  1.52  from  standard  no—  i  =  .0002950  at  F  line. 

The  B  values  of  table  io  show  a  march  to  be  referred  to  temperature  and 
pressure.  So  the  present  unsatisfactory  differences  are  probably  pressure- 
temperature  effects  beyond  the  discrimination  of  the  method. 

One  reason  for  this  discrepancy  which  suggests  itself  is  the  possible  distor- 
tion of  the  glass  plates  at  the  end  of  the  exhaust  tubes  during  the  exhaustion. 
There  may  be  a  residual  temperature  effect  due  to  the  heating  of  the  air  by 
the  beam  which  passes  twice  through  it,  above  the  indicated  temperature  of 
the  surrounding  tube  of  iron.  But  as  Mo-i  is  already  too  large  compared 
with  standard  values,  this  would  make  the  case  worse.  Similarly,  a  larger 
thermal  coefficient  than  the  normal  value  (1/273)  would  further  increase 


REVERSED   AND   NON-REVERSED   SPECTRA. 


131 


the  data  for  /i0—  i.  For  the  case  of  the  F  lines,  the  B  values  found  by  com- 
parison with  the  standard  MQ—  i  (last  column  of  table  12)  might  be  taken  as 
correct  within  the  error  of  method.  Nothing,  however,  has  been  found  to 
account  for  the  correspondingly  large  values  of  .Bo  for  the  C  line. 

75.  Distortion  of  glass  absent. — To  test  the  effect  of  possible  distortion  of 
the  end  plates  of  the  tube,  a  shallow  cell  was  constructed  but  0.8  cm.  deep, 
closed  by  plates  of  the  same  glass.  The  diameter  of  the  tube  was  identical 
with  that  of  the  long  refracting  tubes.  Tests  made  with  the  electric  arc  and 
sodium  line  gave  the  mean  values 


Plenum.  . 
Vacuum  . 


5-25; 


6.6  cm. 
6.4  cm. 


Thus  the  effect  of  exhaustion  is  0.0002 1  cm.    The  long  tube  gave,  on  the  aver- 
age, 0.039  cm-  f°r  *38  cm.  of  length.    Hence  the  air  effect  should  be 

0.039X0.8/138  =  0.00022 


which  is  practically  identical  with  the  value  found, 
ceptible  distortion  referable  to  the  glass  plates. 


Hence  there  is  no  per- 


TABLE  13. — Dispersion  of  air.    D  and  F  lines.  Single  Tube,  length  138  cm.   Barom.,  75.40 
at  28°.    £=74  cm.    Temp.  20°.    St.  refers  to  standard  data.    A  =  po—  i. 


AN0 

St. 

Mean 

St.  A 

to—  I 

Line. 

i&AN 

IO"  

e 

io145o 

XX  i  o6 

loM 

loM 

io"B0,F 

io"B<,,D 

cm. 

cm. 

F 

D 

39-4 
•*8  2 

312.8 
I.OT.  8 

2.24 

F 
io6  Diff.  from  St. 

48.61 

295-0 

304.0 
—  9.0 

2.IO 

19.2 

F 

39-o 

310.1 

1.38 

D 
F 
D 

38-3 
39-0 
38.3 

304.6 
310.5 
304-6 

1.49 

D 
io6  Diff.  from  St. 
lo'Diff.  FandD 

58.93 

292.7 

2-3 

299.4 
-6.7 
4.6 

I.78 
1.83 

2.06 
'2.06 

76.  Further  observations  with  sunlight. — In  the  absence  of  other  than 
inferential  reasons  to  account  for  the  difficulties  met  with,  a  final  series  of 
observations  was  made  between  the  D  and  F  lines  and  a  single  tube,  with 
the  results  given  in  table  13.  The  mean  value  of  B0  found  directly,  viz,  1.70 
Xio~14,  would  be  admissible;  but  the  corresponding  values  of  ^o— i  as  com- 
pared with  the  standard  values  are  again  too  large  and  worse  than  above. 
The  same  is  true  of  the  values  of  BO  found  by  comparison  of  AN/e  with  the 
standard  values  of  /u0—  i,  and  their  coefficients  come  out  differently  for  the 
C  and  F  lines.  In  fact,  the  discrepancy  of  /to— i  is  now  about  3  per  cent, 
whereas  observations  for  &N/e  should  not  be  in  error  more  than  (2  X 1/400  = 
0.0025)  0.5  per  cent.  There  is  thus  something  variable  at  the  limit  of  appli- 
cation of  the  present  method  which  has  persistently  escaped  detection.  I 
have  thought  that  a  distortion  associated  with  the  form  of  the  interference 
pattern  in  passing  from  C  to  the  F  line  may  be  in  question,  as  the  discrepancy 


132  REVERSED  AND   NON-REVERSED   SPECTRA. 

varies  in  different,  otherwise  satisfactory  experiments;  or  the  failure  to  com- 
pletely exhaust  the  tube  may  leave  a  small  error  which  becomes  appreciable 
in  J9. 

77.  Conclusion.  —  If  allowance  is  made  for  the  fact  that  AN"  at  the 
micrometer  is  measured  for  air,  at  barometric  pressure  p'  and  absolute  tem- 
perature T,  the  equation  for  /JLO—  i  at  normal  conditions  would  be 


~~$  1*1$  i-p'AN/ep      X* 

where  the  correction  factor  i—ANp'/pe  would  not  appreciably  modify  the 
results. 

It  is  difficult  to  see,  therefore,  why  the  promising  results  of  §4,  which  are 
quite  as  near  the  standard  data  as  the  method  warrants,  did  not  bear  consis- 
tent fruit  in  the  sequel.  The  direct  values  of  B0X  io14  are  usually  too  small, 
sometimes  too  large,  and  range  from  i  to  2  .  On  the  other  hand,  HQ  —  i  usually 
conies  out  too  large,  whereas  it  should  be  correct  to  a  few  tenths  percentage. 
None  of  the  causes  examined,  temperature,  pressure,  thermal  coefficient, 
flexure  of  glass,  etc.,  quite  account  for  such  a  result.  If  BoX  io14  is  computed 
from  standard  results  for  ju0—  i  and  observed  at  different  spectrum  lines,  the 
data  are  nearly  correct  for  some  lines,  but  too  large  for  other  lines,  so  that  a 
single  constant  does  not  reduce  the  series.  It  does  not  seem  probable,  however, 
that  equation  (i)  is  inadequate;  for  the  results  obtained  with  equal  care  at  dif- 
ferent times  for  the  same  MO—  i  or  B0  are  not  in  accord.  The  discrepancies,  in 
other  words,  are  not  persistent  in  value  and  are  therefore  due  to  some  inci- 
dental cause  which  has  not  been  detected.  It  has  seemed  to  me  that  the  change 
in  shape  of  the  interference  pattern  on  passing  from  red  to  violet,  which  in 
case  of  ordinary  glass  mirrors  is  marked,  may  be  responsible  for  some  of  the 
difficulties  encountered.  This  pattern,  which  for  optically  flat  surfaces  would 
remain  elliptical,  becomes  more  and  more  irregular  as  the  distances,  e,  of  the 
mirror  and  grating  are  increased.  The  distorted  image  shrinks  laterally  from 
red  to  violet  fully  one-half,  so  that  it  is  not  certain  that  the  center  of  figure 
is  actually  a  fiducial  point.  The  question,  however,  would  have  to  be  tested. 


CHAPTER  XI. 


THE  CHANGE  OF  THE  REFRACTION  OF  AIR  WITH  TEMPERATURE. 

78.  Apparatus. — In  the  earlier  report  (Carnegie  Inst.  Wash.  Pub.  149,  III, 
Chap.  15,  p.  223)  I  began  some  experiments  on  the  change  of  the  refrac- 
tive index  of  air  with  rise  of  temperature.  The  question  is  interesting,  inas- 
much as  the  temperature  coefficient  has  not  in  most  investigations  been 
found  identical  with  the  coefficient  of  expansion  of  air,  as  Lorentz  had  obtained 
it  and  as  would  otherwise  be  anticipated;  but  a  value,  over  3  per  cent  larger, 
first  put  forward  by  Mascart,  seems  preferable.  My  earlier  work  was  left 
unfinished,  however,  because  the  design  of  the  apparatus,  in  which  the  refrac- 
tion tube  was  heated  in  an  independent  annular  steam-bath,  was  unsatis- 
factory. It  seemed  to  be  impossible  to  reach  the  temperature  of  the  steam 
in  that  way,  even  after  half  a  day's  waiting.  In  the  present  work,  therefore, 
the  apparatus  is  modified,  so  that  the  steam  may  play  directly  on  the  long 
refraction  tube.  In  this  way  the  temperature  difficulty  was  quite  eliminated. 


93 


J 


The  tube  containing  the  air  column  was  made  of  inch  brass  gas-pipe,  71.7 
cm.  long  (between  windows)  and  2.5  cm.  in  internal  diameter  (A,  fig.  93, 
which  shows  one  end  of  the  apparatus) .  The  ends  were  closed  with  the  usual 
brass  caps  a,  in  which  round  windows,  about  2  cm.  in  diameter,  had  been  cut 
on  the  lathe.  The  ends  were  closed  by  plates  of  glass  g,  secured  between  two 
jackets  of  rubber  and  "  vulcanized"  fiber.  L  shows  the  axis  of  the  beam  of 
light. 

BB  is  the  steam  chamber,  steam  entering  at  5  and  leaving  by  a  similar  tube 
at  the  other  end  of  the  apparatus.  Steam  is  thus  directly  in  contact  with  the 
tube.  The  projecting  end  of  A  is  inclosed  by  a  recess  packed  with  wadding, 

133 


134 


THE   INTERFEROMETRY   OF 


CC.  As  the  walls  of  the  brass  pipe  were  thick  and  the  ends  relatively  short, 
there  seemed  to  be  no  objection  to  this  arrangement.  Care  was  taken  to 
conduct  the  escape  steam  and  hot  gases  away  from  the  interferometer. 

The  displacement  interferometer  was  of  the  linear  type  described  above, 
the  mirrors  M  and  N  and  the  grating  G  being  attached  directly  to  the  wall  of 
the  pier  and  without  an  intervening  rail.  Unfortunately  the  pier  in  a  large 
city  is  also  in  incessant  vibration,  so  that  the  interference  patterns  quiver. 
It  is  this  insuperable  difficulty  which  has  prevented  me  from  reaching  results 
as  accurate  as  were  anticipated.  A  few  of  the  data,  however,  will  be  added 
as  an  example  of  the  efficiency  of  the  method. 

TABLE  14. — Refraction  of  air  at  different  temperatures.    Tube,  71.7  cm.  long, 
2.5  cm.  in  diameter. 


Barometer. 

Temp. 

P- 

lo'XA* 

Barometer. 

Temp. 

P- 

xo'XA* 

77.04  cm. 

19-7° 

75-3 

19-3 

75.12  cm. 

19.9° 

74-3 

19-3 

22° 

19.4 

20° 

19.1 

I 

19-3 

VI 

19.0 

19.2 

76.25  cm. 

21.7° 

75-5 

19.1 

20° 

19-3 

76.98  cm. 

100.4° 

75-5 

15-0 

II 

19.2 

18.7° 

15.0 

19.4 

VII 

15-0 

19.4 

19-3 

77.22  cm. 

19.1° 

75-7 

19.8 

76.55  cm. 

100.2° 

75-3 

15-0 

20° 

19.7 

15-0 

IV 

19.8 

Ill 

15-4 

19.7 

19.7 

76.25  cm. 

22.2° 

75-5 

19.1 

75.12  cm. 

99-7° 

74-3 

15.0 

20° 

19.2 

20° 

15-0 

V 

19.4 

VIII 

14.9 

19.6 

15-0 

19-5 

I5-I 

79.  Observations.— The  data  are  given  in  table  14,  where  the  temperature 
and  barometric  pressure  are  shown  in  the  first  column,  the  differences  in  the 
pressure  p  between  the  plenum  of  air  and  the  exhausted  air  in  the  refraction 
tube  in  the  second  column,  while  the  third  shows  the  values  of  AAT,  the  dis- 
placement of  the  micrometer  corresponding  to  p,  as  found  in  successive  inde- 
pendent experiments  at  the  temperature  given.  For  such  long  distances 
between  grating  and  mirrors  the  ellipses  are  visually  distorted,  and  much 
depends  on  finding  a  satisfactory  sharp  interference  pattern.  This  was  the 
case,  except  in  series  3  and  5,  when  for  incidental  reasons  (outside  tremors) 
the  patterns  were  disagreeably  flickering.  The  observations  are  usually  for 
room  air,  as  the  special  drying  of  air  in  series  3  and  5  made  no  perceptible 
hfference.  At  100°  care  must  be  taken  to  obviate  convection  currents  of 
air,  so  far  as  possible.  The  endeavor  was  made  to  keep  p  as  nearly  as  possi- 
ble at^the  same  value,  apart  from  the  barometer  pressure,  which  does  not 
enter  into  the  equations.  In  series  4  the  values  of  AAT  are  relatively  large, 


REVERSED   AND   NON-REVERSED   SPECTRA.  135 

but  quite  consistent  with  each  other.  The  reason  for  this  could  not  be  made 
out.  But  for  the  inevitable  tremors  the  observations  would  all  have  been 
acceptable. 

80.  Computation.  —  Since  the  ends  of  the  air-tube  are  perpendicular  to 
the  beam  of  light 

(I)  AW 


where  AAT  is  the  difference  of  the  displacements  of  the  micrometer  in  the 
presence  and  absence  of  air  in  the  tube,  e  the  effective  length  of  the  air  column, 
and  j«  the  index  of  refraction  of  the  air  for  the  given  wave-length  X.  The 
equation  presupposes  a  knowledge  of  the  dispersion  of  air  dn/d\;  but,  as  this 
is  small,  the  term  may  be  temporarily  omitted.  If  X  is  constant,  it  corre- 
sponds to  a  constant  correction  of  AAf  throughout  the  experiments. 

Again,  if  we  have  an  equation  of  the  form  of  Mascart's,  /Z76  referring  to 
o°  C.  and  normal  barometer,  and  NQ  to  the  absence  of  air  in  the  tube, 


-i  =p 


-i     76        i  +  erf          Nn-N,    AA/7, 
where  a  and  /3  are  two  constants.     If  the  tube  is  not  quite  exhausted  (SB 
remaining),  the  observations  for  a  plenum  (barometric  pressure,  B)  and 
exhausted  air  being  made  at  the  same  temperature, 

AfrB-W 
__ 

or  nearly 


B  NB-N9 

Thus  if  one  neglects  the  small  correction  i  —  ftB  of  dB 

(3)  AAfc-AW,-*^ 

the  micrometer  displacement  AA/£  in  case  of  complete  exhaustion  at  the 
barometric  height,  B,  and  the  displacement  &NB-.SB  corresponding  to  partial 
exhaustion  B  —  8B,  are  proportional  to  those  pressures.  Since  dB  was  quite 
small,  this  equation  was  assumed,  and  p—  8B  is  thus  nearly  the  height  of  the 
mercury  column  of  the  partially  exhausted  tube.  In  the  table  this  is  briefly 
called  p,  and  differs  from  the  barometric  height. 

Finally,  for  two  partial  pressures  p  and  p'  and  temperatures  t  and  t'  of  the 


AAT  and  AJV'  being  the  micrometer  displacement  corresponding  to  p,  t,  and 
p',  t',  respectively.    Hence  if  care  be  taken  to  make  p  =  p',  nearly, 

i  +  at       «(*'  —  <)    =adt 


AJV    "  AN'  ~AN-&N'Nd 


136  THE   INTERFEROMETRY   OF 

if,  for  brevity,  t'-t  =  8t  and  AN-  AN'  =  dN;  or 


If  p  is  not  quite  equal  to  p',  P(p'—p)  may  still  be  neglected,  but  AN/p  and 
AN'/pr  must  replace  AN  and  AAT',  orAN(i-8p/p)  replace  AAT  where  5£  = 
£-/. 

On  applying  equation  (5)  to  series  i,  2,  3,  for  which  p  is  nearly  constant, 

$t  =  'jg.$0  5Af=  0.00423  01  =  0.00380 

applying  it  to  series  4,  5,  7,  similarly, 

§*=79.8°  SN=  0.00404  01  =  0.00404 

The  mean  value  is  thus  a  =  0.00392.  The  reason  of  this  difference  is  found  in 
series  4,  where  AN"  is  excessive.  In  fact,  if  we  compare  percentage  errors  of 
a  and  8N 

da/a  AN  N 

= 


so  that  an  error  of  5  per  cent  in  dN  would  be  an  error  of  over  5  per  cent  in  a. 
For  the  case  where  the  fringes  tremble  this  is  inevitable.  If  the  mounting  were 
without  tremor,  however,  dN  should  be  guaranteed  to  sXio'5  cm.,  corre- 
sponding to  the  evanescence  of  a  single  interference  ring,  so  that  a  should  be 
determinable  to  i  per  cent,  even  in  case  of  a  tube  of  the  length  71.7  cm.  given. 
If  tt  is  small  or  *'  small,  equation  (5)  becomes,  approximately, 


This  equation  may  be  used  to  find  the  successive  values  of  AN  in  the  table, 
if  the  second,  for  instance,  is  supposed  to  be  correct.  It  appears  that  the 
first  and  fifth  differ  about  equally  (±0.0001  cm.)  from  the  second,  but  the 
error  of  the  fourth  (—0.00028)  is  excessive.  Hence  if  this  second  datum  be 
taken  as  the  mean  of  series  1,2,5,  and  combined  with  the  two  data  for  100°, 

AAT=  19.28  SN  -0.421  52  =  78.6°  t'=  100.3°  a  =  o.oo38s 

This  is  the  more  probable  result  of  table  14  and  would  agree  with  Mascart's 
value,  0.00382. 

Somewhat  later,  the  independent  series  of  observations  6  and  8  were  carried 
out.  The  interference  pattern  at  99.7°  was  exceptionally  quiet  and  clean,  but 
at  lower  temperatures  this  was  not  better  than  usual.  The  results  are 


4.i5  5*=  79. 

somewhat  below  the  preceding  value. 


81.  Final  experiments  at  100°.—  Somewhat  later,  at  a  time  when  the  labo- 
ratory was  relatively  quiet  and  after  the  same  effective  improvements  had 
been  made  in  the  mounting  of  the  interferometer  mirrors,  the  experiments 


REVERSED   AND   NON-REVERSED   SPECTRA. 


137 


at  100°  were  repeated.  The  optical  measurements  were  satisfactory,  or  at 
least  just  short  of  the  counting  of  interference  rings  for  measurement.  The 
arc  lamp,  moreover,  which  is  unsteady,  would  scarcely  suffice  for  this  purpose. 
The  results  obtained  were  as  follows  (table  15) : 

TABLE  15. — Refraction  of  air  at  different  temperatures. 


Bar. 

Temp. 

P 

icfaN 

Bar. 

Temp. 

P 

IO'A#' 

75.6    cm. 

21.8° 

74.0 

19-3 

76.72  cm. 

100.3° 

74.0 

I5-I 

19.2 

20.5° 

154 

19-3 

15.0 

19.1 

I5-I 

76.65  cm. 

21.8° 

74.0 

19.6 

15-5 

19.4 

15.0 

19-5 

15-2 

19.2 

154 

- 

19-3 

15-2 

15-3 

15-2 

78.5° 


If  the  mean  values  of  AW  and  AW'  be  taken  and  a  computed 
io3AW=ig.22  io3AW'=is.22  6W=4.oo  8t-- 

the  result  is 

01  =  0.00361 

As  these  experiments  were  the  smoothest  and  were  made  under  the  most 
satisfactory  conditions,  they  are  probably  the  most  trustworthy.  I  have  not, 
therefore,  been  able  to  obtain  evidence  for  a  value  of  a  (between  o°  and  100°) 
greater  than  the  coefficient  of  expansion  of  gases,  though  it  must  be  confessed 
that  the  method  in  its  present  surroundings  is  not  sufficiently  sensitive  to 
furnish  a  definite  criterion. 

Later  results  at  low  temperatures  (series  3)  like  the  above  series  4,  table 
14,  again  gave  a  high  result  for  AW,  in  each  case  consistently.  It  is  probable 
that  the  interference  pattern  changes  between  the  case  of  a  plenum  and  of 
highly  exhausted  air,  owing  either  to  flexure  of  the  glass  ends  or  to  some  other 
cause,  or  possibly  depending  only  on  the  form  of  the  pattern  which  happens 
to  appear.  In  such  a  case  the  lines  of  symmetry  for  W  (plenum)  and  W  (ex- 
haustion) would  differ,  introducing  a  systematic  error  very  difficult  to  obviate. 
Thus  different  values  of  AW  often  follow  a  difference  of  adjustment  of  the 
mirror  at  the  micrometer,  while  all  cases  for  the  same  adjustment  are  practi- 
cally identical. 

82.  Experiments  at  red  heat. — To  investigate  the  feasibility  of  such  experi- 
ments, an  inch  steel  tube  (bicycle  tube),  68  cm.  long,  with  flanges  brazed  on 
at  the  ends,  and  an  exhaustion  tube  near  the  middle,  was  heated  in  an  organic 
combustion  furnace  to  low  red  heat.  The  ends  just  projected  outside  the 
furnace  and  were  closed  by  plate-glass  windows  with  a  jacket  of  asbestos 
between  (applied  wet  and  dried);  or,  finally,  with  a  jacket  of  aluminum 
cement,  clay,  plaster,  etc.  These  short  but  relatively  cold  ends  are,  of  course, 


138  THE   INTERFEROMETRY   OF 

an  objection  to  the  method,  but  no  better  device  was  found.  Even  so,  the 
windows  frequently  cracked  and  had  to  be  replaced.  Such  an  apparatus 
naturally  leaks,  particularly  at  low  temperatures,  where  the  viscosity  of  air 
is  relatively  small,  so  that  the  experiments  as  a  whole  are  merely  tentative. 
To  maintain  the  exhaustion  as  high  as  70  cm.,  it  was  necessary  to  keep  the 
air-pump  at  work.  To  reduce  this  annoyance  the  exhaustions  were  at  first 
not  carried  above  60  cm.  of  mercury.  With  the  interference  fringes,  however, 
no  serious  difficulty  was  experienced  after  the  tube  had  taken  definite  shape. 
Distortion  of  fringes  was  inevitable,  but  centers  of  symmetry  for  measure- 
ment were  always  available. 

The  first  experiments  were  made  without  exhaustion,  at  low  and  high 
temperature  (low  red  heat).  The  difference  of  displacement  8N  between 
cold  (2  5°)  and  hot  was  (for  instance)  in  two  different  experiments 

25°  ioW  =35.0  cm.        35.2  cm. 

red  hot        ioW=a8.s  28.6 

or  ioz5N=  6.5  6.6 

at  atmospheric  pressure.  The  8N  so  obtained  makes  no  allowance  for  the 
change  of  refractive  index  of  the  hot  glass  ends,  nor  for  any  displacement  or 
rotation  or  warping  of  the  ends  during  the  course  of  the  experiment,  which 
required  a  lapse  of  an  hour  or  two. 

In  the  next  experiment,  therefore,  the  method  of  exhaustion  was  attempted, 
the  partial  vacuum  used  being  about  16.6  cm.  when  the  full  barometer  read 
76.64  cm.  Thus  p  =  6o  cm.  An  example  of  the  results  obtained  is  given  in 
the  following  data. 

Cold  Tube. 

Pressure  76.6  cm.        ioW   =34.8  cm.        34.7  cm. 

Pressure  16.6  22.7  22.5 

P=  6o.O  IO3AAT=I2.I  12.2 

Red-hot  Tube. 

Pressure  76.6  cm.        ioW   =24.6  24.5  26.8  26.0  cm. 

Pressure  16.6                             20.1  19.4  20.0  20.0 

P  =          60.                 io3AA/"      4.5  5.1  6.8          6.0 

In  the  two  experiments  at  the  end  readjustment  was  necessary,  as  the  red-hot 
tube  warped  during  the  exhaustion.  In  the  last  case  the  glass  cracked.  The 
first  two  data  should  therefore  be  taken,  so  that 

io3AAr=i2.icm.        io3AAr'  =  4.8cm.        ioW=7.3  cm.       £  =  25°  £  =  6o  cm. 
If  equation  (5)  above  is  solved  for  t'  the  result  is 

or  if  a=  1/273 

This  result  is  certainly  small,  as  one  would  estimate  the  temperature  (red 
heat)  at  several  hundred  degrees  higher.  Unfortunately  the  relatively  cold 


REVERSED   AND   NON-REVERSED   SPECTRA.  139 

ends  of  the  tube  and  the  leakage  at  the  windows  both  contribute  to  a  low 
value  of  t'  '.  But  these  do  not  seem  to  be  adequate  reasons.  It  is  more  probable 
that  the  longitudinal  radiation  of  the  air  on  the  one  hand  and  the  value  of 
i  /a  =  2  73  assumed  (if  this  is  too  small)  may  be  the  chief  causes  for  the  low 
value  of  t'.  It  is  not,  of  course,  possible  to  come  to  any  further  decision;  but 
the  experiments  are  distinctly  unfavorable  to  the  large  value  of  a  (small 
i  /a)  above  considered. 

The  method  is  not  adapted  for  very  high  temperatures,  since  equation  (7) 
may  be  written 


and  therefore,  since  r'AAT  =  rA/V, 


where  (r  referring  to  absolute  temperature)  AJV'  rapidly  reaches  the  limit  of 
accurate  measurement. 

83.  Further  experiments  at  high  temperatures.  —  A  variety  of  experiments 
were  now  made  to  obtain  a  more  nearly  tight  joint  at  the  ends,  by  using 
various  clays,  aluminum,  etc.,  as  cements,  but 
without  success.  Finally,  an  improvement  was 
obtained  by  using  plaster  of  paris  in  the  way 
shown  in  figure  94.  A  is  the  end  of  the  hot 
tube  in  the  combustion  furnace  F.  The  flange  / 
is  set  somewhat  back,  so  that  packing  of  plaster 
p  may  secure  the  window  g  to  the  end  of  the 
tube.  The  plaster  is  put  on  wet  and  allowed  to 
dry  thoroughly.  Lying  outside  of  the  furnace,  94 
it  is  never  heated  to  redness.  The  joint  is  at 
first  fairly  good,  though  it  gradually  deteriorates  at  high  temperatures,  and 
must  be  replaced.  In  this  way  the  following  results  were  found: 

TABLE  16. 

p.     io*&N                               p.     lo'XAW  p.  lo'AAT 

Just  below  red  heat    74.5     12.4    Cold  tube  (22°)     74.0     17.6  Low  red  heat    73.5     8.9 

12.5                                             17.7  8-5 

12.0                                               17.8  8.6 

18.1  8.8 

17.8 

Thus,  from  the  first  and  second  series,  t'=  154°;  from  the  first  and  third  series, 
''  =  330°.  As  in  the  first  experiments  tried,  both  of  these  data  are  much  too 
low.  Here  they  can  hardly  be  referred  to  the  leak,  since  this  was  smaller. 
The  ends  are  exposed  not  more  than  i  or  2  cm.  each,  or  a  total  length  of  about 
70  cm.  of  tube. 

Some  adjustment  is  needed  at  the  mirrors,  to  place  the  slit  images  in  coinci- 
dence for  the  case  of  an  exhaustion,  as  compared  with  a  plenum  of  air.  This 
adjustment  is  slight,  but  unfortunately  its  effect  on  AN'  can  not  be  estimated. 


140  THE   INTERFEROMETRY  OF 

Cooling  of  gas  as  resulting  from  longitudinal  radiation  might  be  suggested, 
but,  as  it  was  not  encountered  in  the  case  of  the  steam  tube,  it  would  not 
seem  to  be  menacing  here. 

Finally,  it  will  be  seen  from  equation  (8)  that  the  effect  of  a  leak  is  to  make 
AN'  too  small.  It  will  be  larger  as  the  vacuum  is  more  perfect.  Hence  tr 
should  be  too  large  for  this  reason.  A  small  t'  can  not  be  due  to  a  leak.  The 
exhaustion  effect,  since  the  gas  expands  into  a  vacuum,  can  not  be  serious. 
None  of  these  incidental  difficulties  seem  adequate  to  account  for  the  large 
temperature  discrepancies  consistently  obtained.  All  things  considered,  it 
seems  to  me  most  probable  that  the  temperature  coefficient,  as  the  gas  enters 
the  region  of  red  heat  more  fully,  continually  decreases,  and  that  this  is  the 
real  explanation  of  the  low  temperature  values  obtained. 

The  apparatus  was  now  taken  apart  and  provided  with  a  fresh  jacket. 
After  drying,  the  cold  apparatus  again  appeared  in  good  condition.  The 
results  with  the  barometer  at  75.55  cm.  were 

Cold  (22°)  p        io3AAT  Red  hot  p        icMW 

73.0            17.7  73.0               8.0 

17-8  

17.8 

Unfortunately  the  glass  cracked  after  the  first  experiment  at  red  heat. 
The  data  for  AW  (cold)  agree  almost  exactly  with  the  preceding  results.  The 
high  temperature  would  be  t' =383°,  again  enormously  too  low.  Nevertheless, 
if  the  values  of  a  were  in  question,  as  the  temperature  must  have  been  at 
least  850°,  this  would  come  out  as  low  as  01=0.0015.  The  misgivings  already 
enumerated  apply  here  as  before.  As  the  experiments  are  very  laborious 
they  were  abandoned  at  this  point,  for  it  did  not  seem  that  further  work 
would  materially  enhance  the  result;  nor  was  it  thought  necessary  to  actually 
measure  the  high  temperatures. 

84.  Flames. — In  the  earlier  report  on  the  refraction  of  flames  an  abnormally 
low  result  of  n  was  obtained  for  the  ignited  gases.  I  have  since  repeated  this 
work  with  additional  improvements.  It  appears  that  it  is  quite  possible  to 
look  through  the  peak  of  the  blue  case  (symmetrically)  without  destroying 
the  interference  pattern  as  a  whole,  though  this  naturally  quivers  excessively. 
The  last  of  the  new  results  showed  for  the  presence  (A/7)  and  (A/")  of  the  flame 
the  micrometer  readings : 

N',  flame    0.029      0.029      0.029      0.029       0.029 
N,  air  .0297       .0295       .0294       .0296       .0296 

Hence  the  mean  difference  is  0.00056  cm.,  or  per  centimeter  of  breadth 
(2.3  cm.), 

8N  =  0.00024  cm. 

If  the  space  occupied  by  flames  were  vacuum,  the  difference  would  have  been 
0.000268  per  linear  centimeter.  Thus  AW' =  0.00002  cm.,  which  lies  within 
the  error  of  observation,  but  is  otherwise  quite  of  the  order  to  be  expected 
for  the  hot  gases  in  question. 


REVERSED  AND   NON-REVERSED   SPECTRA.  141 

85.  Conclusion. — Though  the  experiments  made  are  of  a  tentative  charac- 
ter, the  inference  seems  warranted  that,  so  far  as  my  work  goes,  the  tem- 
perature coefficient  a  of  air  at  low  temperature  is  identical  with  the  coeffi- 
cient of  expansion  of  gases.  At  high  temperatures  the  value  of  a  seems  to 
decrease  rapidly,  in  proportion  as  the  gas  is  more  highly  ionized  at  red  heat. 

It  has  occurred  to  me  that  such  ionization  might  load  the  gas  in  relation 
to  the  light-wave  passing  through  it,  and  that  the  observed  excess  of  index 
of  refraction  over  the  value  anticipated  at  high  temperatures  might  be 
explained  in  this  way.  But  air  ionized  by  the  X-rays  shows  no  such  effect. 
Neither  does  the  refraction  of  flames  at  high  temperatures,  so  far  as  can  be 
made  out,  show  a  large  value  of  the  refractive  index  of  the  ignited  gases. 

It  is  difficult  to  see  how  the  experiment  at  red  heat  can  be  improved,  unless 
a  quartz  tube  is  made  for  the  purpose.  But  even  here  the  difficulty  of  obtain- 
ing adequately  plane  parallel  ends  and  a  tube  of  sufficient  breadth  is  formid- 
able. The  attempt  to  grind  in  reentrant  glass  cylinder-like  stoppers  at  the 
end  of  the  tube  was  thought  of,  but  did  not  succeed. 


CHAPTER  XII. 


ADIABATIC  EXPANSION  OBSERVED  WITH  THE  INTERFEROMETER. 

86.  Introductory. — In  the  preceding  report1  I  tested  a  number  of  receivers 
in  which  air  was  expanded  adiabatically,  by  passing  one  of  the  component 
beams  of  the  displacement  interferometer  through  the  air  contained.  The 
vessels  then  used  were  not  very  satisfactory,  being,  as  a  rule,  not  long  or 
capacious  enough  to  insure  trustworthy  results.  Moreover,  the  interferometer 
did  not  at  that  time  admit  of  the  introduction  of  long  or  bulky  apparatus, 
whereas  in  the  new  form  a  length  of  almost  150  cm.  is  available.  The  main 
purpose  of  the  research  will  thus  be  to  ascertain  how  long  and  thin  a  tube  may 
be  made  to  be  serviceable  for  expansion  experiments.  Furthermore,  it  seemed 
worth  while  to  repeat  the  work  preliminarily  with  a  large,  staunch  tank  since 
found  in  the  laboratory.  This  was  a  heavy  cylinder  of  cast  brass,  about  27.1 
cm.  (inside)  and  closed  by  plates  of  heavy  glass,  each  0.56  cm.  thick  and  20.3 
cm.  apart  (inside),  the  whole  containing  a  volume  of  air  of  about  11,713 
cubic  centimeters,  to  be  increased  to  12,800  cubic  centimeters,  because  of  the 
efflux  pipe.  The  expansion  pipe  was  2  inches  in  diameter  and  closed  by  a 
2>£-  inch  brass  stopcock,  with  a  plug  practically  floating  in  oil  to  prevent  the 
ingress  of  air  from  without.  The  glass  plates  were  secured  by  iron  bolts,  a 
layer  of  resinous  cement  (equal  parts  of  beeswax  and  resin)  between  glass  and 
the  flat  end  faces  of  the  cylinder  being  introduced  to  prevent  leakage. 

To  expand  the  gas  in  the  receiver,  the  2 -inch  pipe  communicated  with  a 
tall,  galvanized  iron  boiler  used  as  a  vacuum  chamber,  29.4  cm.  in  diameter 
and  147  cm.  high,  thus  containing  a  volume  of  99,800  cubic  centimeters,  or 
100,200  cubic  centimeters  with  the  influx  pipe.  It  was  in  communication 
with  a  large  air-pump  and  provided  with  a  mercury  gage  for  the  measurement 
of  the  partial  vacuum  produced  by  the  pump.  The  air  flowing  into  the  air- 
chamber  after  exhaustion  was  dried  in  the  usual  way  and  the  influx  controlled 
by  a  fine  screw  stopcock.  There  was  a  special  opening  for  a  thermometer. 
Vacuum  and  air-chamber  were  rigidly  connected  by  a  brass  union  with  a 
rubber  washer.  There  was  no  appreciable  leakage  so  far  as  the  atmosphere 
without  was  concerned.  The  2-inch  stopcock,  however,  was  not  quite  tight 
within,  so  that  air  passed  very  slowly  from  the  air  to  the  vacuum  chamber, 
in  proportion  as  their  pressures  were  different;  but  as  the  air-chamber  is  in 
service,  either  at  atmospheric  pressure  (the  influx  cock  being  open)  or,  after 
exhaustion,  at  approximately  the  same  pressure  as  the  vacuum  chamber,  this 
leakage  was  of  no  appreciable  consequence.  Otherwise  the  interference  pat- 
tern would  not  have  been  stationary. 

While  this  apparatus  was  not  long  enough  to  fully  realize  the  advantages 
of  the  method  of  displacement  interf erometry  for  the  purposes  in  question, 
Carnegie  Inst.  Wash.  Pub.  149,  Part  II,  Chapter  IX,  1912. 


REVERSED   AND   NON-REVERSED   SPECTRA. 


143 


it  was  useful  for  testing  the  ring  method  in  comparison  with  the  former.  The 
equivalent  of  a  vanishing  interference  ring  is  here  not  immediately  given  in 
terms  of  the  wave-length  of  light,  since  the  rings  move  through  the  spectrum. 

With  the  exception  of  a  few  incidental  experiments  of  my  own,  optic 
methods  of  the  present  kind  have  not  hitherto  been  used.  They  are  here  par- 
ticularly applicable,  since  the  number  of  the  rings  vanishing  in  a  given  region 
of  the  spectrum  has  merely  to  be  counted  after  the  sudden  exhaustion  and 
during  the  period  of  slow  influx  of  air. 

Succeeding  parts  of  the  chapter  will  refer  to  other  available  forms  of  ap- 
paratus with  similar  ends  in  view,  and  the  additional  purpose  of  ascertaining 
how  long  and  narrow  an  apparatus  may  be  shaped,  without  seriously  inter- 
fering with  the  adiabatic  measurements;  for  if  the  apparatus  is  increased 
indefinitely  in  length  and  diameter,  it  is  obvious  that  the  suddenness  of  the 
exhaustion  through  any  available  pipe  will  be  more  and  more  impaired.  The 
same  is  true  if  the  apparatus,  for  a  given  (sufficient)  length,  is  too  narrow, 
though  for  a  different  reason. 

TABLE  17. — Values  of  7.    Bulky  air  chamber,   F=99,8oo  cub.  cm.,  »=  11,620  cub.  cm. 
=  1.116.    C=952.6;  1+^=1.0341;  0=20.3  cm- 


Series. 

t 

Po 

P 

10'AJV 

J 

Number 
of  rings. 

7' 

I 

°c. 
22.4 

cm. 
75-88 

cm. 
56.38 
56.88 

cm. 
I-I5 
•95 

02 

I.I5 

1-39 
1  44. 

30 
29 

2Q 

1.49 
1.50 

I  5O 

1.02 

Ill 

29 

1.50 

II 

22.4 

75-88 

47-68 

1.53 

i-33 
1-45 
i-47 

1.29 

i-53 
1.38 
1.36 

46 
46 
46 
46 

1.58 

1.58 

1.58 

1-53 

III 

22.8 
21.  2 

75-70 

38.50 

1-95 

2  25 

1.38 

14. 

'6l 

'do 

1-54 

I  54. 

23-6 
24.0 



2.05 
1.91 

.29 

.42 

61 
62 

1.53 
1.50 

IV 

222.5 
222.5 
23-6 
21.6 

76.75 

30-35 

2.65 

2.2O 
2-73 

2.J.Q 

.28 
•65 

.21 

I.VJ 

77 
76 
78 
78 

1.58 

i-59 
i-54 
i.sj. 

1  Count  broken  owing  to  flicker  of  arc ;  obtained  from  rhythm.    2  Sunlight. 

87.  Experiments  with  short,  bulky  air-chambers. — An  example  of  the  data 
obtained  is  given  in  table  17,  where  the  ratio  of  specific  heats,  7,  computed 
directly  both  from  displacement  of  ellipses  and  7'  from  interference  rings,  is 
shown  in  detail.  The  original  pressure  of  the  air-chamber  is  that  of  the  barom- 
eter, pQ.  The  pressure  of  the  vacuum  chamber  is  given  under  p.  The  dis- 
placement, A7V,  from  four  independent  observations  in  each  case  and  the 
number  of  interference  rings  vanishing  from  exhaustion  to  plenum  are  the 
data  chiefly  of  interest.  It  has  not  been  possible,  according  to  the  table,  to 
*  Carnegie  Inst.  Wash.  Pub.  149,  Part  II,  §  83,  9.129;  §  85,  p.  135.  1912. 


144  THE   INTERFEROMETRY   OF 

place  the  micrometer  with  an  accuracy  of  more  than  0.0002  cm.  or  0.0003 
cm.  in  successive  cases,  AN  being  the  difference  of  two  readings,  each  uncer- 
tain to  10^  cm.  But  the  effect  of  this  is  to  throw  out  7  by  about  the  same 
number  of  tenths,  so  that  the  roughness  of  values  in  the  table  is  inevitable. 
On  the  other  hand,  however,  7  obtained  by  displacement  is  usually  too  small, 
whereas  the  value  computed  from  the  evanescence  of  rings  is  always  much 
too  large.  Thus  in  the  first  series  there  should  have  been  an  evanescence  of  3  1 
rings,  in  the  second  of  about  50  rings,  in  the  third  of  64  rings,  in  the  fourth  of 
85  rings.  The  reason  for  this  discrepancy  is  very  hard  to  determine,  but  will 
be  considered  in  the  next  paragraph.  The  mean  values  of  7  from  displace- 
ment and  from  rings  are  usually  more  nearly  correct  than  either,  as  if  the 
errors  were  equal  and  opposite  in  the  two  cases.  The  error  is,  in  some  way 
which  has  not  been  made  out,  associated  with  the  placing  of  the  micrometer. 
Thus,  without  apparent  cause,  the  micrometer  reading  with  a  plenum  of  air 
may  differ  by  several  10"*  cm.,  so  that  if  these  discrepancies  are  in  opposite 
directions  the  value  of  7  shows  such  large  divergences  as  in  series  4,  for  in- 
stance. In  other  words,  the  error  appears  to  be  extraneous  to  the  method  of 
experiment. 

It  has  been  suggested  that  the  number  of  vanishing  rings  observed  is  approx- 
imately about  10  per  cent  too  small  throughout,  and  that  the  corresponding 
data  for  7,  though  excessive,  are  nevertheless  of  the  same  order  of  value. 
Experiments  were  made  to  determine  whether  the  change  of  wave-length,  X, 
influenced  this  result.  This  was  done  by  allowing  the  center  of  ellipses  in 
one  case  to  move  from  the  D  line  towards  the  red,  in  the  other  from  the  yellow 
into  the  D  line.  The  mean  wave-length  would  in  the  last  case  be  smaller,  and 
one  may  estimate  the  former  as 

-XD  AAT 


where  AN0  is  the  displacement  of  mirror  which  passes  the  center  of  ellipses 
from  the  C  to  the  D  line.    This  was  found  to  be  io3AAT0  =  28.1  cm.    Hence 

AAT 
- 


Even  in  the  final  case,  therefore,  where  io3A]V=2.s,  \D  would  not  be  in  error 
by  more  than  0.5  per  cent.  Using  sunlight  and  at 

£0  =  76.75  cm.  £  =  48.65  cm.  7  =  294.5°  (abs.  temp.) 

the  number  of  rings  R  were  counted  when  the  ellipses  traveled  into  the  D 
line  and  from  the  D  line,  respectively,  with  results  of  which  the  following 
are  examples  : 

From  D  line,  tf  =  47     46     46  Mean  #  =  46.3 

Into  D  line,    #  =  45     47     46.5     46     Mean  #  =  46.1 
Indifferent,     R  =  46     46  Mean  £  =  46.0 

These  results  agree  with  the  second  series  of  table  17,  and  there  is  thus  no 
appreciable  difference. 


REVERSED  AND   NON-REVERSED   SPECTRA.  145 

One  may  note  that  the  results  for  7,  when  rings  are  counted,  are  consistently 
too  large,  but  always  of  the  same  order.  In  fact,  if  R  were  increased  by  the  re- 
duction factor  r076/2  73^0,  the  values  of  7  would  all  be  nearly  correct;  but  there 
is  no  reason  for  such  a  correction.  Moreover,  since  the  data  for  7  obtained 
from  AN  (ellipses  brought  back  to  fiducial  position)  and  from  R  (ellipses  dis- 
placed) are  each  separately  consistent  with  each  other,  the  discrepancy  can 
not  be  due  to  leakages  of  air,  as  these  would  affect  both  measurements  in  the 
same  way.  The  only  source  of  error  which  is  not  common  to  both  (apart  from 
the  displacement  of  ellipses)  is  the  possible  distortion  of  the  glass  upon  exhaus- 
tion; for,  in  case  of  AN,  measurement  is  made  at  a  plenum  and  at  maximum 
exhaustion  only,  but  at  varying  pressures  for  the  case  of  rings.  Thus  if  the 
rings  needed  are  supposed  to  increase  in  the  ratio  of 

i.3     1.6     2.0    2.5 

roughly,  an  approximate  adjustment  of  the  two  sets  of  observations  would 
also  be  obtained.  Moreover,  the  effect  of  flexure  would  be  an  increase  of  the 
path  of  the  beam  in  glass  and  so  counteract  the  negative  effect  of  decreased 
density. 

88.  Effect  of  strained  glass.  —  To  detect  the  possible  effect  of  the  inward 
flexure  of  the  two  plates  of  glass,  a  metallic  ring  about  25  cm.  in  internal 
diameter  was  provided.  To  this,  two  glass  plates  of  about  the  same  thickness 
(0.8  cm.  each)  as  in  the  above  vessel  were  cemented  free  from  leakage  and 
kept  in  place  by  clamps.  The  distance  apart  of  the  two  plates  within  was 
but  1.8  cm.,  so  that  the  micrometer  displacement  due  to  exhaustion  of  air 
was  reduced  to  a  small  value.  Hence,  if  the  flexure  of  the  glass  plates  due  to 
exhaustion  and  the  reverse  were  optically  appreciable,  it  should  here  be 
detected. 

To  compute  the  residual  air  effect  for  the  lamella  of  air,  e—  1.8  cm.  thick, 
we  may  write 

(1)  Ctfo  =  p/(M-i)=A/U-i) 

where  C=  952.6,  t?0  is  the  temperature  of  the  isothermal  experiment,  M  and 
ju0  the  index  of  refraction  of  air  at  the  pressures  p  and  po.  Furthermore, 

(2)  n—i=fJo—i—AN/e 

if  AN  is  the  micrometer  displacement  for  the  pressure  difference  p—po  at  t?o- 
Finally,  if  n  is  the  number  of  rings  vanishing  or  of  fringes  passing  at  the 
sodium  line,  then 


Thus  ilp—p0=Bp,  then 


58.93   Ctto  ^ 


146 


THE   INTERFEROMETRY   OF 


so  that  n  and  dp  are  proportional  quantities.     The  following  results  were 
found : 


sp 

No.  of 
rings 
observed. 

No.  of 
rings 
computed. 

Middle  of  glass  plate  .  . 

4  cm.  above  middle..  .  . 
8  cm.  above  middle..  .  . 
At  edge 

[30  cm. 
45 
[60 
45 
45 
45 

7.2 

10.0 

13.2 

9.8 

10.0 

10.5 

6-5 
9-7 
13.0 

9-7 
9-7 
9-7 

The  observed  data  are  the  means  of  5  or  6  trials.  As  it  is  difficult  to  observe 
the  rings  without  interruption  in  an  agitated  laboratory,  there  is  no  doubt 
that  observed  and  computed  values  are  coincident.  The  first  and  last  rings 
are  not  easily  counted,  and  individual  data  were  found  to  agree  with  the 
computed  results  perfectly.  Finally,  if  the  glass  strain  were  effective  (for 
there  is  actual  flexure),  it  would  be  shown  in  the  observations  made  by  pass- 
ing the  beam  through  different  parts  of  the  plate  of  glass,  between  the  center 
and  the  edge.  No  consistent  difference  was  found. 

Hence  an  appreciable  strain  effect  is  also  absent,  and  the  reason  for  the 
discrepancy  in  the  two  sets  of  values  from  AA7  and  from  n,  in  table  17,  remains 
outstanding. 


In  the  preceding  report  *  the  equation  for  the  value  of  7 
log  f  /ft 


89.  Equations. 

is  deduced  as 


or 


Here  p0  and  p  are  the  pressures  in  the  air-chamber  (barometric)  and  the 
vacuum  chamber  respectively,  before  exhaustion,  t?0  the  original  tempera- 
ture of  the  air,  AAT  the  displacement  of  the  micrometer  corresponding  to  the 
shift  of  the  ellipses  on  exhaustion.  If  the  air-chamber  is  quite  tight,  A7V  may 
be  taken  at  any  time.  C  and  i  +x  are  the  optic  constants 

C  =  ft/#o(/io  —  i)  =952.6 

for  dry  air,  being  the  optic  gas  constant,  if  MO—  i  replaces  p0,  the  normal 
density  of  the  gas.  To  allow  for  the  dispersion  of  air  an  empirical  equation 
(convenient  in  the  present  calculation), 


was  constructed.  The  deduction  assumes  that  the  centers  of  ellipses  are 
brought  back  again  to  the  fiducial  line  D,  of  the  spectrum,  the  micrometer 
displacement  in  question  being  AAT. 

*  Carnegie  Inst.  Wash.  Pub.  149,  Part  n,  pp.  166-168. 


REVERSED   AND   NON-REVERSED   SPECTRA.  147 

In  the  case  where  rings  are  counted,  however,  the  center  of  ellipses  leaves 
the  D  line  by  a  short  distance,  less  than  one-tenth  of  the  interval  between  the 
C  and  D  lines.  In  such  a  case,  if  n*  =  A*+bJ\z  for  air  and  ng=  Ag+bg/\* 
for  glass,  the  micrometer  displacement  to  bring  the  ellipses  back  again  from 
X'  to  X  should  be 


e&  and  eg  being  the  lengths  of  air  and  glass*  in  the  beam.    Here 

<?a  =  2o.3cm.         6a  =  icr14Xi.65         ea&.  =  icr14X33.5 
es=  2     cm.        6g  =  io-12X    48        <?g&e  =  icr12X96 

so  that  the  effect  of  air,  where  &a  is  variable  with  pressure,  is  but  0.3  per  cent 
of  the  glass  effect  and  may  in  the  first  approximation  be  neglected.  The 
equation  may  therefore  be  written  : 


~       X3      3^,  X        6egbe 

If  the  mean  data  from  series  I  be  inserted  (5AT  =  96oXio-6  when  X  refers  to 
the  D  line) 

5X_96oXio-6Xio-sXo.3473_T  n_.Vn  co 
T-  6X96X10-^"       ~10    X°'58 


For  the  case  of  the  C  and  D  lines  5X/X  =  3.35/58.9  =  o.o57,  roughly,  about  ten 
times  the  preceding  distance. 

In  fact,  the  observations  made  for  the  estimate  given  in  the  preceding  para- 
graph (semi-displacement), 

AX    .      .  , 


compared  with  the  present 


576X10-" 


_ 


are  quantities  of  the  same  order,  though  one  would  have  expected  closer 
coincidence. 

The  discrepancy  observed  between  the  method  of  measurement  in  terms 
of  the  displacement  (AAT  to  bring  the  ellipses  back  to  the  fiducial  position) 
and  the  method  of  counting  rings  can  not,  therefore,  be  explained  as  the 
result  of  a  change  of  wave-length  X  in  the  latter  case;  i.e.,  the  equation 


where  HQ  —  H  is  the  number  of  vanishing  rings  of  the  mean  wave-length  X,  is 
at  fault  for  some  other  reason.    Curiously  enough,  the  ring  method  is  essen- 

tially simple,  as  it  reduces  to  7  =  ,  —  (T\*  ^  n<>  an(^  n  are  ^ne  numDer  of  rings 
*  Thickness  of  glass  plates  of  air-chamber,  1.3  cm.;  of  the  plate  of  the  grating,  0.7  cm. 


148  THE   INTERFEROMETRY  OF 

vanishing  when  a  plenum  of  air  and  the  adiabatically  exhausted  air,  respec- 
tively, are  introduced  into  one  of  the  beams.    Since 

/*>  —  i  =  n<>X/2e = po/Cdo 
this  is  equivalent  to 

AAT=(n.-«)X/2 

90.  Experiments  with  long  tubes.  Diameter,  one  inch.— The  difficulty 
encountered  in  the  case  of  the  preceding  experiments  was  the  small  value  of 
the  displacement  AAT  obtained.  As  a  consequence,  every  little  incidental 
disturbance  produced  a  large  effect  in  7.  It  is  the  purpose  of  the  present 
experiments  to  remedy  this  defect  by  using  long  tubes  by  which  AN  may  be 
increased  over  ten  times.  It  was  particularly  of  interest,  moreover,  to  begin 
with  relatively  thin  tubes,  and  inch  gas-pipe  suggested  itself  for  the  purpose. 
The  value  of  7  to  be  expected  will  necessarily  be  too  small,  as  the  air  must 
undergo  reheating  before  the  exhaust  cock  can  be  closed.  The  question,  how- 
ever, is  whether  consistent  values  of  7  will  be  found,  even  for  these  extreme 
conditions  and  for  large  variations  of  pressure.  Obviously  the  window  plates 
will  not  produce  discrepancies,  as  has  been  directly  shown  in  paragraph  88. 

The  gas-pipe  installed  was  143.4  cm.  long  within.  To  make  the  junction 
with  the  vacuum  chamber,  a  straight  pipe  of  the  same  diameter  and  about 
75  cm.  long  was  needed  between  the  main  pipe  and  the  2^-inch  stopcock. 
The  connecting  pipe,  together  with  the  tube  itself,  is  probably  the  chief  cause 
of  the  resistance  to  flow  and  the  low  value  of  7  found,  but  it  was  not  possible 
to  shorten  it. 

The  large  stopcock  inevitably  leaked  slightly  when  the  pressures  were 
different  in  the  two  chambers;  but  immediately  after  exhaustion  this  made 
no  appreciable  difference,  as  the  two  pressures  are  then  nearly  the  same.  In 
fact,  no  rings  vanish  from  the  spectrum  from  this  cause.  Just  before  exhaus- 
tion, however,  after  closing  the  gas-pipe  by  the  fine  influx  stopcock,  appreciable 
leakage  is  shown  by  the  spectrum.  Hence  the  exhaustion  must  be  made 
immediately  after  the  influx  cock  is  closed.  Some  low  results  at  the  outset 
are  referable  to  this  difficulty. 

The  tube  was,  as  usual,  filled  with  dry  air  after  exhaustion.  The  results 
are  given  in  table  18,  in  the  same  way  as  in  the  preceding  case.  The  experi- 
ments themselves  were  throughout  satisfactory,  no  difficulty  being  encountered 
at  the  interferometer.  The  work,  moreover,  is  equally  trustworthy  at  low 
and  at  high  exhaustions,  a  result  which  is  rather  surprising.  In  the  latter 
case,  as  the  total  displacement,  AAT,  is  over  0.0276  cm.,  the  7  contained  should 
be  correct  within  i  per  cent. 

Only  one  attempt  was  made  to  find  AAT  by  the  march  of  the  interference 
fringes.  Fully  276  were  observed,  and  it  is  here  necessary  to  count  the  fringes 
passing  the  D  line,  since  the  ellipses  are  displaced  throughout  the  greater  part 
of  the  length  of  the  spectrum;  but  this  introduces  no  inconvenience  whatever. 
The  difficulty  is  due  to  the  time  needed  in  counting  so  many  evanescences; 
for  dunng  this  interval  the  electric  lamp  is  liable  to  flicker  seriouslv,  or  some 


REVERSED   AND   NON-REVERSED   SPECTRA. 


149 


commotion  will  occur  in  the  laboratory  or  without,  tending  to  make  the  count 
uncertain.  The  rings  disappear  temporarily  during  the  tremor.  In  a  quiet 
laboratory,  however,  and  with  sunlight  replacing  the  arc  light,  this  would  be  a 
method  of  precision.  Thus,  for  instance,  at  the  highest  exhaustions  used,  over 
900  fringes  would  have  to  pass  the  D  line,  a  datum  from  which  7  could  be 
accurately  obtained. 

TABLE  1 8. — Values  of  7.    Iron  gas-pipe,  i  inch  internal  diameter.    €=952.6. 
e=  143.4  cm.     i+x=i.034i. 


Series. 

1 

Po 

P 

IO*AN 

y 

No.  of 
rings. 

y 

I 

°c. 
19.2 

cm. 

76.84 

cm. 
57-84 

cm. 

8-75 
8.60 

8-35 
8.40 

.18 

.21 

•24 
.24 

276 

1.28 

II 

19.1 

76.28 

48.08 

12.95 
i"\  I  s 

.20 
.18 

1  3  IS 

.18 

.... 



13-25 

I  -3  10 

•17 
10 



13-25 

•17 

III 

19.2 

76.28 

38.98 

18.05 

18.10 

17.95 

17.80 

.14 
.14 
.15 

.17 

IV 

19-3 

76.28 

29.78 

22.70 
22.80 

•15 

.14 

22  6S 

.IS 

22.80 

.14 

V 

19-3 

76.28 

20.88 

27.85 
27.60 
27.60 

.12 
.14 
.14. 

If  we  compare  the  mean  results  for  7  with  the  exhaustion  used  (pressure  p 
in  the  vacuum  chamber,  full  barometric  pressure  p0  in  the  air-chamber),  the 
results  decrease  slightly  as  the  vacuum  is  higher.  Thus 


If         p=    58cm. 
Then  7=     1.23 


48  cm. 
1.18 


39  cm. 


30  cm. 
1.14 


21  cm. 
1-13 


which  is  what  might  have  been  expected,  except  that  the  rate  of  decrease  is 
much  less  than  would  be  surmised.  There  seems  thus  to  be  no  objection  to 
the  use  of  high  exhaustions,  which  in  turn  give  a  better  value  of  7  from  the 
large  range  of  A  A/"  obtained. 

The  low  mean  value  of  7  obtained  has  been  referred  to  the  resistance  of 
the  inch  piping  to  the  outflow  of  air.  It  is  probably  not  due  to  the  stop- 
cock, as  incidental  differences  in  the  speed  of  opening  and  closing  would 
otherwise  have  shown  a  marked  effect.  One  may  conclude  that  the  air  in 
the  long  inch  gas-pipe  expands  adiabatically  with  a  coefficient  7  between  r.i 
and  1.2,  in  case  of  such  exhaustions  as  the  above. 


150 


THE   INTERFEROMETRY   OF 


91.  The  same.  Diameter  of  tube,  two  inches. — The  experiments  were  now 
continued  by  enlarging  the  diameter  of  the  tube  to  2  inches.  Brass  gas-pipe, 
1.35  cm.  long,  to  be  closed  with  thick  glass  plates,  was  at  hand.  To  connect 
the  same  with  the  vacuum  chamber,  a  similar  2-inch  pipe,  115  cm.  long,  as  far 
as  the  2^4-inch  stopcock,  was  necessary.  Moreover,  as  this  was  in  the  way  of 
the  light  received  from  the  grating,  the  beam  was  reflected  by  an  offset  con- 
sisting of  two  silver  mirrors  in  parallel.  No  difficulty  was  found  with  this 
arrangement,  and  the  sodium  line  was  in  view  to  give  evidence  if  any  acci- 
dental displacement  should  occur. 

Unfortunately,  the  ellipses  obtained  were  somewhat  irregular  open  forms 
(i.e.,  half  ellipses),  and  the  endeavor  to  secure  small  closed  patterns  did  not 
succeed.  This  annoyance  depending  chiefly  on  the  parts  of  the  mirror  and 
grating  used,  and  on  shifting  accessories,  is  not  easily  controlled.  The  indi- 
vidual measurements  of  AN  are  therefore  not  as  good  as  those  recorded  in 
table  1 8,  where  a  displacement  of  jo"4  cm.  was  assured.  They  suffice,  how- 
ever, for  the  present  purposes. 

The  new  data  are  given  in  table  19,  t  being  the  temperature  of  both  cham- 
bers, po  the  initial  normal  pressure  of  the  air-chamber  (2-inch  pipe),  and  p 
that  of  the  vacuum  chamber. 

TABLE  19. — Values  of  7.     Brass  gas-pipe,  2  inches  internal  diameter. 
C=952.6;  i+x=     1.0341.    £=135.3  cm.    (V+v)/v=  1.049. 


Series. 

t 

Po 

P 

lo'AJV 

y 

I 

°C. 
17.0 

cm. 
75.66 

cm. 
56.46 

cm. 
7-15 
7-35 
7-25 

7-55 

1-35 
1.30 

1-33 
1.27 

II 

17.0 

75-66 

47-36 

10.85 

IO  OS 

1-33 

I    ^1 

II.  10 

11.05 

1.29 
1.30 

III 

16.1 

75-86 

38.36 

14-93 
15-15 
15-10 
15.38 

i-3i 
1.28 
1.29 
1.26 

IV 

16.2 

75-86 

29.36 

19.53 
19-57 
19.27 

19-75 

1-25 
1-25 
1.28 
1.23 

V 

16.4 

75-86 

20.36 

23.71 
23-95 
23-70 
23.75 

1.27 

1-25 
1.27 
1.26 

The  effective  value  of  y  in  these  experiments  is,  for  the  lower  exhaustions, 
above  7=1-3,  showing  a  considerable  improvement  over  the  data  for  the  inch 
tube,  which  were  not  much  above  7=1.1.  This  was  to  be  inferred,  of  course ; 
but  it  was  not  expected  that  the  increment  of  7  due  to  increased  diameter 


REVERSED   AND   NON-REVERSED   SPECTRA. 


151 


would  be  so  rapid.  It  would  seem  to  be  probable,  therefore,  that  if  a  4-inch 
tube  were  used  the  conditions  for  obtaining  a  trustworthy  value  of  7  would 
be  nearly  met. 

As  the  exhaustions  in  a  successive  series  are  gradually  increased  (initial 
partial  vacua  from  £  =  56.46  cm.  to  £  =  20.36  cm.  in  the  vacuum  chamber), 
the  observed  values  of  y  gradually  but  slowly  decrease,  the  mean  values  being 


cm.  to  75.9  cm.) 

£  =  56.46 
7=   1.32 


38.36 
1.29 


29.36 
1.25 


20.36  cm. 
1.26 


where  the  fourth  value  is  too  small,  for  incidental  reasons.  This  general 
result  is  also  to  be  expected;  but  it  is  rather  remarkable  that  with  such  high 
exhaustions  as  those  finally  used  the  decrease  of  7  is  not  more  marked. 

The  work,  as  a  whole,  progressed  smoothly  throughout,  the  only  interfer- 
ence with  precision  being  the  incidental  occurrence  of  open  ellipses.  To  obtain 
other  patterns  would  have  required  longer  additional  adjustment  than  the 
work  at  the  present  stage  seemed  to  warrant. 

92.  The  same.  Diameter  of  tube,  four  inches. — The  first  experiments  made 
with  the  4-inch  tube  are  given  in  table  20.  The  completed  apparatus  showed 
a  slight  leak,  which  could  not  be  detected  after  long  searching.  The  tube 
was  therefore  admitted  for  a  tentative  series  of  experiments.  The  exhaust 
pipe  here,  as  above,  was  rigid  and  straight,  but  only  2  inches  in  diameter, 
with  a  2X-incn  stopcock.  To  exhaust  the  air-chamber,  the  handle  of  the 
cock  was  suddenly  jerked  over  an  angle  180°  between  the  two  closed  positions. 
The  plug  virtually  floated  in  oil,  as  shown  elsewhere. 

TABLE  20. — Values  of  y.    Brass  pipe,  4  inches  internal  diameter. 
C=952. 6;  i+x=  1.0341;    «=  126.9.    (V-\-v)/V=i.ng.    Small  leak  in  apparatus. 


Series. 

t 

Po 

P 

10'  'AN 

•y 

°C. 

cm. 

cm. 

cm. 

I 

19.9 

76.15 

57-00 

5-90 

1.42 

6.10 

1.37 

6  2\ 

1.34 

6  IS 

I  -?6 

II 

23.0 

75-79 

38.39 

12.40 

1.36 

12.73 

1.32 

12.60 

1-30 

12-75 

I-3I 

III 

23-2 

75-79 

2939 

15.50 

1.40 

1571 

i-37 

15.77 

1.37 

As  a  whole,  the  results  are  disappointing;  and  they  are  irregular,  for  mean 
readings  could  not  be  made  because  of  the  leak.  They  are,  nevertheless, 
interesting,  inasmuch  as  with  some  of  the  above  data  they  point  out  a  special 
source  of  discrepancy.  It  will  be  seen  that  the  7  values  tend  to  decrease  in 
successive  measurements,  beginning  with  a  high  value,  which  is  here  nearly 


152 


THE   INTERFEROMETRY  OF 


correct.  This  can  not  be  referred  to  the  temperature  of  the  4-inch  tube, 
because  the  initial  optic  density  is  necessarily  measured.  It  must  therefore 
be  due  to  the  temperature  of  the  vacuum  chamber.  It  follows,  therefore, 
that  the  time  allowed  in  these  experiments,  between  observations,  though 
sufficient  for  establishing  the  initial  temperature  of  the  air-chamber,  is  not 
sufficient  for  the  much  larger  vacuum  chamber.  The  two  chambers  are  thus 
no  longer  at  the  same  temperature,  a  condition  which  the  equations  implicitly 
assume. 

The  apparatus  was  now  taken  apart  and  thoroughly  overhauled.  After 
reassembling  the  parts,  the  chamber  was  found  free  from  leakage.  As  the 
exhaust  pipe  was  in  the  way  of  the  beam  of  light  entering  the  telescope,  the 
offset,  consisting  of  two  parallel  mirrors  firmly  adjusted,  was  used  without 
annoyance,  here  as  above.  The  work  throughout  progressed  smoothly,  though 
the  ellipses  were  again  not  as  satisfactory  in  form  as  would  have  been  desirable. 

TABLE  21. — Values  of  7.    Data  as  in  Table  n,  4"  brass  pipe. 


Series. 

/ 

P° 

P 

lO'Atf 

•y 

I 

°c. 

16.5 

cm. 
75-90 

cm. 
56.80 

cm. 
6-34 
6.15 
6.27 
6.  to 

1-33 
1-37 
1-34 
I  ^5 

II 

16.6 

75-90 

47.60 

9.25 

Q  4.O 

1.38 

I  16 

Q    7Q 

I  ^4. 

16.7 

76.19 

47.89 

9.20 

9-55 
9-3i 
9-45 

1-39 
1-33 
1-37 
i-35 

III 

16.9 

76.19 

38.69 

12.87 

12  70 

1-34 
i  16 

12  80 

I  35 

12  76 

I  ^5 

IV 

17.0 

76.19 

29.69 

16.26 

•35 

16  25 

•28 

V 

17.1 

76.19 

20.69 

20.00 
19.60 

19-73 
19.69 

•35 

.40 
•38 
•39 

Table  21  contains  the  results.  Changes  in  the  values  of  AAT  in  a  given 
series  are  most  likely  referable  to  the  form  of  the  interference  pattern,  indi- 
rectly to  the  flickering  of  the  electric  lamp.  There  seems  to  be  no  evidence 
to  associate  them  with  the  manner  in  which  the  2^-inch  stopcock  is  opened 
and  closed.  This  was  merely  jerked  around  180°,  between  the  two  closed 
positions  of  the  plug,  and,  so  far  as  can  be  seen,  the  rate  of  motion  is  adequate. 
The  successive  observations  show  no  consistent  difference,  as  was  the  case 


REVERSED   AND   NON-REVERSED   SPECTRA. 


153 


in  the  preceding  table.  Hence  this  discrepancy  has  been  eliminated.  What 
is  most  interesting  is  that  the  4-inch  tube  shows  no  consistent  difference  in 
the  7  values  for  high  or  low  exhaustion.  Thus  the  mean  values  under  increas- 
ing exhaustion,  p,  are 

£  =  56.8      47.6      38.7       29.7       20.7 
7  =   1.35       1.36       1.35       1.36       1.38 

Accidentally  the  highest  value  of  7  belongs  to  the  highest  exhaustion. 

The  chief  anticipation  of  the  work  (i.e.,  that  with  a  4-inch  tube  the  true 
value  of  7  would  appear)  has  not  been  fulfilled.  The  value  obtained  is  still 
much  below  normal,  successive  results  ranging  as  follows: 

Diameter  of  tube 2.5       5.0       10.0  cm. 

Mean  7 1.17     1.29       1.36001. 

Diameter  of  exhaust  pipe    2.5       5.0        5.0  cm. 

The  relatively  small  increase  between  the  tubes  5  cm.  and  10  cm.  in  diam- 
eter is  disappointing.  At  the  rate  obtained  from  the  first  two  experiments 
(see  fig.  95)  a  3-inch  tube  should  have  been  nearly  sufficient.  At  the  rate 
established  by  the  last  two  observations,  however,  a  tube  at  least  5.5  inches 


U 
1* 

1-0 

t 

_u-  — 

I 

/ 

*^~ 

^—  < 

I  < 
Sbioun 

..,wwA« 

s  —  #- 

95 

in  diameter  would  be  needed  to  obtain  trustworthy  values  of  7.  These  differ- 
ences are  possibly  due  to  the  exhaust  pipe,  which  in  case  of  the  last  observation 
does  not  increase  in  size.  Hence  a  3-inch  pipe  with  a  4-inch  stopcock  may 
be  estimated  as  being  adequate  for  7  measurement,  provided  the  exhaust  pipe 
is  straight  and  clear  throughout. 

The  observations  were  broken  off  at  this  point,  with  the  object  of  searching 
for  some  means  of  obtaining  a  more  sensitive  and  regular  interference  pattern. 
If  the  method  is  to  be  ultimately  successful,  then  icr4  cm.  on  the  micrometer 
must  be  guaranteed.  If  the  ellipses  are  not  quite  regular  or  not  closed,  this 
is  not  the  case.  A  more  sensitive  method  of  defining  optic  density  is  thus  in 
question. 


CHAPTER  XIII. 


MISCELLANEOUS  EXPERIMENTS. 

93.  Effect  of  ionization  on  the  refraction  of  a  gas.  —  It  seemed  interesting  to 
test  this  question  carefully,  although  a  negative  result  was  to  be  expected. 
Accordingly  one  component  beam  was  surrounded  by  a  thick  iron  tube,  while 
the  other  was  allowed  to  travel  freely  in  air,  along  a  path  energized  by  the 
X-rays.  For  this  purpose  the  X-ray  bulb  was  placed  near  the  grating  and  the 
radiation  directed  toward  the  mirror  N,  the  beam  GM  being  inclosed.  A  thick 
sheet  of  lead,  i  foot  square,  was  placed  behind  the  bulb  to  additionally  screen 
off  radiation  along  GM.  Under  these  circumstances  the  ionization  along  GN 
must  have  been  enormous  by  comparison  with  GM.  Quiet  ellipses  were  pro- 
duced in  the  interferometer,  and  the  effect  of  opening  the  X-ray  current  and 
closing  it  again,  alternately,  was  observed.  Not  the  slightest  deformation  of 
the  ellipses  or  any  motion  of  the  fringes  could  be  detected.  An  ionization  effect 
is  therefore  wholly  absent.  It  might  have  been  supposed,  for  instance,  that 
the  ions  present  might  load  the  wave  of  light  and  produce  an  appreciable  result 
in  the  interferometer  (cf.  fig.  92). 

Since  a  shift  of  o.i  of  a  ring  would  probably  have  been  detected,  AAT  = 
0.000005  cm.  would  have  produced  a  perceptible  effect.  Hence,  since  n—i 
is,  roughly,  equal  to  A  N/e,  the  value  of  the  ionization  effect  could  not  exceed 


The  ionization  effect  can  not,  therefore,  exceed  o.oi  per  cent  of  ju~  i- 

To  further  test  this  question,  the  iron  tube,  i  inch  in  diameter  and  138  cm. 
long,  was  provided  with  a  fine  axial  wire  about  0.02  cm.  in  diameter,  passing 
through  central  holes  in  the  glass  plates  at  the  end.  The  ends  of  the  wire  were 
drawn  tight  by  hard-rubber  rods  on  the  outside,  so  that  the  tube  became 
a  cylindrical  condenser.  All  holes  were  sealed  hermetically  with  resinous 
cement.  The  interference  fringes  were  clearly  producible. 

The  poles  of  an  induction  coil  were  now  connected  with  the  inner  wire  and 
the  tube,  respectively,  to  alternately  change  the  condenser  and  discharge  it, 
with  the  object  of  strongly  ionizing  the  air  within.  On  partial  exhaustion  the 
whole  tube  became  luminous,  on  account  of  the  discharge,  in  the  usual  way. 
The  best  results  were  obtained  with  a  plenum  of  air  when  but  two  storage 
cells  actuated  the  coil.  Under  these  circumstances  no  sparks  passed  from  core 
to  shell  of  the  iron  condenser  tube,  while  the  air  within  was  intensely  ionized 
by  the  silent  discharge.  On  closing  the  current,  from  0.5  to  i  per  cent  of  the 
rings  was  swept  inward  at  once.  On  opening  it,  the  rings  again  emerged.  This 
inward  motion,  however,  was  in  the  same  sense  as  the  effect  of  a  decrease  of 
154 


REVERSED   AND   NON-REVERSED   SPECTRA.  155 

density,  such  as  would  result,  for  instance,  from  rise  of  temperature  or  from 
partial  exhaustion.  Hence  the  effect  observed,  though  very  definite,  would 
correspond  to  a  temperature  effect  due  to  electrical  currents  traversing  the  air. 
One  should  expect  the  effect  of  ionization,  if  appreciable,  to  be  the  reverse  of 
this.  With  voltages  high  enough  to  produce  sparks  in  the  tube,  the  inter- 
ference figures  naturally  show  violent  agitation  or  quiver.  If  the  displacement 
in  question  is  one  ring  and  5  denotes  differences,  S(AAT)  =  30X10-*  cm. 

If  only  temperature  changes,  one  may  write,  roughly,   A./V.r  =  constant, 
r  referring  to  absolute  temperature,  whence 


40  X  10 

if  results  found  for  a  similar  tube,  above,  be  taken. 

Thus  ?>T  =  2.2Xicr7  degrees  centigrade  is  the  average  temperature  incre- 
ment, for  the  whole  length  of  the  tube. 

When  but  a  single  cell  was  used  to  energize  the  coil,  no  effect  could  be  recog- 
nized. In  case  of  two  cells,  moreover,  when  the  plenum  of  air  was  replaced 
by  a  partial  vacuum  of  i  cm.  or  less,  so  that  an  arc  was  seen,  no  effect  was 
observable,  although  the  reddish  light  colored  the  field  of  the  telescope. 

There  are  two  points  of  view,  however,  from  which  the  assumption  of  a  tem- 
perature effect  is  not  admissible.  If  the  pipe  is  closed,  so  that  the  density  of 
the  air  contained  remains  unchanged,  there  is  no  difference  in  the  phenomenon. 
But  there  should  not,  for  the  case  of  constant  density,  be  any  effect,  unless  the 
nature  of  the  gas  is  changed.  Again,  the  effect  is  instantaneous  and  not 
increased  on  keeping  the  circuit  closed.  The  simple  explanation  in  terms  of 
temperature  made  above  must  therefore  be  taken  with  reservation.  At  all 
events,  the  effect  of  ionization  would  be  small  and  equivalent  to  a  dilution  of 
the  gas  of  but 

---  X  2.2  X  io~7  or  about  icr9 
273 

of  its  density,  when  sparks  are  about  to  occur. 

94.  Mach's  interferences.  —  It  is  frequently  necessary  to  use  the  interferom- 
eter in  such  a  way  that  but  one  ray  passes  in  a  given  direction  ;  i.e.,  the  rays  are 
not  to  retrace  their  path.  Interferometers  of  this  kind  are  treated  above,  but 
Mach's  design  offers  advantages,  which  will  be  presently  pointed  out.  As  a 
rule,  in  using  these  interferometers,  the  center  of  the  elliptic  interference  pat- 
tern is  remote  and  the  lines  are  hair-like  and  found  with  great  difficulty.  These 
annoyances  are  overcome  when  the  apparatus  is  put  together  as  follows: 

In  figure  96,  L  is  the  vertical  sheet  of  light  from  a  collimator  impinging  on 
the  strip  of  plate  glass  gg,  half-silvered  on  one  side,  toward  or  near  the  ends. 
The  pencil  L  is  thus  reflected  to  the  opaque  mirror  N  and  transmitted  to  the 
opaque  mirror  M  (on  a  micrometer),  and  then  reflected  to  the  other  end  g'  of 
the  glass  strip  gg'  .  Thereafter,  both  the  pencils,  Mg'  and  Ng',  are  available; 


156  THE   INTERFEROMETRY   OF 

but  it  is  generally  more  convenient  to  use  the  former  (Afg')»  reflecting  it  from 
the  plane  opaque  mirror  m  to  the  telescope  at  T.  When  L  came  from  sunlight, 
or  from  an  arc  light,  etc.,  the  white  images  of  the  slit  were  very  bright.  After 
putting  them  in  coincidence,  horizontally  and  vertically,  by  aid  of  the  three 
adjustment  screws  on  the  mirror  M,  Ives  prism-grating  G  may  be  placed  in 
front  of  the  objective  of  the  telescope.  A  very  brilliant  spectrum  thus  appears, 
and  the  fringes  are  easily  found  by  moving  the  micrometer  slide  which  carries 
M  to  the  proper  position.  In  my  apparatus  gg'  was  about  50  cm.  long  and 
gM=gN  about  2  meters.  The  telescope  is  sufficiently  near  M  to  manipulate 
the  micrometer,  the  mirror  m  being  so  placed  that  the  beam  just  misses  the 
strip  gg'. 

ctif 


96 


The  interference  pattern,  found  at  once  and  satisfactorily  centered,  consisted 
of  large,  broad  circles.  On  moving  the  micrometer  M  from  evanescence  on  one 
side  of  the  center  to  evanescence  on  the  other,  the  slide  was  found  to  have 
moved  over  about  2  mm.  With  a  stronger  telescope  to  magnify  the  fine,  hair- 
like  fringes,  this  distance  would  have  been  larger.  It  is  interesting  to  compare 
this  datum  displacement  with  the  datum  found  in  the  case  of  the  phenomenon 
above,  where  a  range  of  over  0.5  cm.  (double  path-difference)  was  observed. 
In  the  present  experiment  the  range  is  smaller,  because  the  interference  pattern 
falls  below  the  limit  of  visibility  before  the  possibility  of  interference  is 
exhausted.  Mg'  slides  along  gr  when  M  moves. 

95.  A  Rowland  spectrometer  for  transmitting  and  reflecting  gratings,  plane 
or  concave. — In  the  above  experiments  I  had  occasion  to  examine  a  variety  of 
gratings,  and  it  was  therefore  desirable  to  devise  a  universal  instrument  by 
which  this  could  be  accomplished  without  delay.  The  method  chosen  is  sim- 
ilar to  that  previously  described,*  but  its  details  have  been  greatly  simplified, 
on  the  one  hand,  and  made  more  generally  applicable,  on  the  other.  It  seems 
permissible,  therefore,  to  give  a  brief  description. 

*  Carnegie  Inst.  Wash.  Pub.  No.  149,  Chapter  I,  1911. 


REVERSED   AND   NON-REVERSED   SPECTRA. 


157 


In  figure  97,  GG'  and  HE'  are  double  slides  like  the  carriage  bed  of  a  lathe, 
each  about  1.5  to  2  meters  long  and  10  cm.  wide,  rigidly  fastened  together. 
They  are  placed  at  right  angles  to  each  other  on  a  flat  table,  the  vacant  distance 
between  G'  and  HH'  being  less  than  a  meter.  For  ordinary  purposes  they  need 
not  be  screwed  down.  A,  B,  D,  K,  are  flat  carriages,  or  tables,  provided  with 
screw  sockets  for  supporting  the  different  standards,  and  capable  of  sliding  to 
and  fro  with  a  minimum  of  friction.  A  carries  the  micrometer  slit  5.  B  and  C 
are  joined  by  the  Rowland  rail  R,  whose  length  is  thus  equal  to  the  radius  of 
the  concave  grating  to  be  examined,  or  nearly  so,  so  that  the  ends  of  R  are  on 
vertical  axes  at  b  and  d.  B  also  supports  the  table  C  (somewhat  enlarged  in 
the  side  elevation,  fig.  98) ,  on  which  the  table  t  of  the  grating  g  may  be  adjusted 
on  its  leveling  screws.  To  secure  a  common  axis,  b,  e,  the  rod  at  ace  is  twice 
bent  at  right  angles.  Moreover,  if  c  is  turned  to  one  side,  the  supporting  rod  e 
may  be  screwed  into  the  vacant  socket  b  at  the  end  of  R.  For  the  case  of  fig- 
ure 98,  the  angle  of  diffraction  6  is  varied  and  \=D  sin  6,  where  D  is  the  grat- 
ing space.  For  the  other  case  (c  being  turned  aside  and  C  screwed  into  and 
turning  with  6)  the  angle  of  incidence  is  varied  and  X  =  D  sin  i.  This  is  much 
simpler  in  form  than  the  early  method  used. 


97 


Finally,  the  table  C  carries  the  essentially  new  addition  to  the  apparatus 
(shown  in  front  elevation  in  fig.  99),  viz,  the  long  slot  ff,  adapted  to  support 
the  right-angled  reflecting  prism  E  and  at  the  same  time  to  allow  free  play  to 
the  rail  R  within  _/f.  Figure  99  then  shows  the  progress  of  the  rays  (turned  90° 
to  the  front  in  a  horizontal  plane)  from  the  slit  or  collimator,  S.  They  are 
doubly  reflected  at  E,  return  in  a  vertical  plane  and  then  impinge  on  the  grat- 
ing at  G.  The  rays  thereafter  pass  along  the  rail  R  (fig.  97)  and  are  examined 
by  a  strong  eyepiece  at  d  (not  shown),  rigidly  but  adjustably  attached  to  the 
near  end  of  the  rail. 

The  displacement  of  K  along  HH'  is  accurately  measurable  on  a  parallel 
scale  with  vernier  (not  shown)  .  If  Xi  and  x%  are  the  two  symmetrical  readings 
on  opposite  sides  of  the  virtual  slit  image  at  5  (fig.  97),  and  R  the  radius  of 
the  concave  grating,  and  x  =  xt  —  Xi 


sin 


,  or  sin 


158  REVERSED   AND   NON-REVERSED    SPECTRA. 

If  a  plane  grating  is  used,  a  weak  lens  L  is  attached  to  the  rail  R  and  moves 
with  it,  so  that  its  focus  is  in  front  of  the  ocular  d  (with  cross-hairs).  In  this 
case  5  is  a  collimator.  If  a  transmitting  grating  is  examined,  the  collimator 
5  (fig.  99),  etc.,  are  merely  to  be  lowered,  and  the  prism  E  is  superfluous.  It 
need  not  even  be  removed.  Naturally,  it  is  in  the  interest  of  accuracy  to  have 
all  the  standards  like  e  and  h  as  short  as  possible. 

Finally,  in  the  equation  ^  =  ~5»  itD=io*d,  the  values  d  and  R  are  usually 

of  the  same  order  (175  cm.)  for  gratings  with  about  15,000  lines  to  the  inch. 
In  this  case  we  may  make  the  rail  length  R  =  d,  whence 

\=X/2 

Even  in  case  of  the  concave  grating,  when  ultimate  precision  is  not  aimed 
at,  some  variation  of  the  distance  SS'  =  2SE,  nearly,  is  admissible  without 
destroying  the  definition.  The  carriage  D  with  the  prism  E  may  be  moved 
fore  and  aft  on  the  slides  GG'  until  the  focus  at  d  is  sharp.  The  values  of  x  are 
usually  of  the  order  of  100  to  125  cm.,  so  that  an  accuracy  of  Angstrom  units 
is  easily  obtainable  without  special  refinement. 


THE  INTERFEROMETRY  OF  REVERSED  AND 
NON-REVERSED  SPECTRA 

PART  II 


By  CARL  BARUS 

Hazard  Professor  of  Physics  and  Dean  of  the  Graduate  Department 
in  Brown  University 


PUBLISHED  BY  THE  CARNEGIE  INSTITUTION  OF  WASHINGTON 
WASHINGTON,  1917 


CARNEGIE  INSTITUTION  OF  WASHINGTON 
PUBLICATION  No.  249,  PART  II 


PRINTED  BY  J.   B.  LIPPINCOTT  COMPANY 

AT  THE  WASHINGTON  SQUARE  PRESS 

PHILADELPHIA,   U.  S.  A. 


CONTENTS. 


CHAPTER  I. — Methods  for  Reversed  and  Non-Reversed  Spectrum  Interferometry. 

PAGE. 

1 .  Introductory 9 

2.  Apparatus.     Figs.  I,  2,  3 9 

3.  Measurements.     First-  and  second-order  spectra.     Tables  1,2.    Figs.  4,  5 II 

4.  Continued.     First-order  spectra 13 

5.  Continued.     Second-order  spectra.    Table  3 14 

6.  Theory.     Fig.  6 14 

7.  Compensator  measurements.     Sharp  wedge 15 

8.  Continuation.    Revolving  plate.    Table  4.    Fig.  7 16 

9.  Continuation.     Air  column.    Tables  5,  6 18 

10.  Continuation.     Babinet  compensator.     Fig.  8 19 

11.  Micrometer  displacement  of  the  second  grating,  G'  (fig.  3).    Table  7 20 

12.  Prism  method.    Reflection.     Table  8.    Figs.  9,  10,  II 21 

13.  Prismatic  refraction.    Figs.  12,  13 25 

14.  Prism  methods  without  grating.     Figs.  14,  15,  16 26 

15.  Displacement  parallel  to  rays.    Table  9 28 

16.  Breadth  of  efficient  wave-fronts.    Apparent  uniformity  of  wave-trains.     Rotation 

of  fringes.     Figs.  17,  18 30 

17.  Film  grating.     Figs.  19,  20 32 

18.  Non-reversed  spectra.     Figs.  21,  22,  23,  24 34 

19.  Non-reversed  spectra.    Restricted  coincidence.     Figs.  25,  26 38 

43 
45 
48 
52 
55 


20.  The  same,  continued.    Homogeneous  light.    Dissimilar  gratings. 

21.  The  same,  continued.     Duplicate  fringes.     Figs.  27,  28,  29. 

22.  The  same.     Prismatic  adjustment.     Figs.  30,  31.. 


23.  Apparent  lengths  of  uniform  wave-trains.    Tables  10,  II.     Figs.  32,  33,  34. 

24.  Normal  displacement  of  mirrors  (rf=o).     Figs.  35,  36,  37 


)•     Figs 

25.  Diffraction  at  M,  N,  replacing  reflection.     Table  12".    Figs.  38,  39 57 

26.  Experiments  with  the  conca 

27.  Polarization.     Figs.  40,  41.. 


26.  Experiments  with  the  concave  grating 59 


CHAPTER  II. — The  Interferences  of  Inverted  Spectra. 

28.  Introductory 62 

29.  Apparatus.     Non-inverted  spectra.     Fig.  42 62 

30.  Apparatus  and  results  for  inverted  spectra.     Figs.  43,  44,  45 64 

31 .  Wave-fronts  narrowed.     Table  13.     Figs.  46,  47,  48 65 

32.  Inverted  spectra.     Further  measurements.     Table  14 67 

33.  Rotation  of  fringes.     Fig.  49 69 

34.  Range  of  displacement  varying  with  orientation  of  reflector  P' 70 

35.  Range  of  displacement  varying  with  dispersion 72 

36.  Spectra  both  reversed  and  inverted.     Figs.  50,  51 73 

37.  Experiments  with  the  concave  grating 74 

38.  Conclusion.     General  methods.     Fig.  52 75 

39.  Displacement  interferometry.     Equations.     Fig.  53 77 

40.  Continued.    Reversed  spectra,  etc.    Tables  15,16 80 

CHAPTER  III. — Elongation  of  Metallic  Tubes  by  Pressure  and  the  Mesurement  of  the 
Bulk  Modulus  by  Displacement  Interferometry, 

41 .  General  method  and  apparatus.     Fig.  54 84 

42.  Remarks  on  the  displacement  interferometer.     Figs.  55,  56 85 

43.  Observations.     Thick  steel  tube 86 

44.  Further  experiments.     Tables  17,  18.    Fig.  57 87 

45.  Brass  tube.     Tables  19,  20.    Figs.  58,  59,  60,  61 90 

46.  Thin  steel  tube.    Table  21 92 

47.  Conclusion.     Thermodynamic  application.     Fig.  62 93 

3 


CONTENTS. 


CHAPTER  IV. — Refr activity  Determined,  Irrespective  of  Form,  by  Displacement  Inter- 
ferometry. 

PAGE. 

48.  Introductory 95 

49.  Preliminary  experiments.    Figs.  63,  64 95 

50.  Apparatus 96 

51.  Equations.    Table  22 97 

52.  Observations.     Tables  23,  24.     Figs.  65,  66 99 

53.  Dispersion  constants.     Tables  25,  26.     Fig.  67 100 

54.  Further  observations.     Tables  27,  28.    Figs.  68,  69 102 

55.  Conclusion.    Table  29 105 

CHAPTER  V. — Displacement  Interferometry  in  Connection  with  U-Tubes.     Jamin's 
Interferometer. 

56.  Introduction.    Figs.  70,  71,  72 107 

57.  Apparatus.     Michelson  interferometer 107 

58.  Equations 109 

59.  Observations no 

60.  Jamin's  interferometer.     Figs.  73,  74,  75,  76 ill 

61.  Vertical  displacement  of  ellipses.     Figs.  77,  78 1 14 

62.  Displacement  interferometer.    Jamin  type.     Tables  30,  31,  32,  33.     Fig.  79  ....  115 

63.  Broad  slit  interferences.   Achromatic  fringes 120 

64.  Wide  slit.     Homogeneous  light.     Sodium  flame.     Figs.  80,  81,  82,  83,  84,  85 121 

65.  Vertical  displacement.    Table  34.    Fig.  86 126 

66.  Angular  displacement  of  fringes.     Table  35.     Fig.  87 128 

CHAPTER  VI. — The  Displacement  Interferometry  of  Small  Angles  and  of  Long  Dis- 
tances.    Complementary  Fringes. 

67.  Parallel  rays  retracing  their  path.     Figs.  88,  89,  90,  91,  92 130 

68.  Groups  of  achromatic  fringes.    Table  36.    Fig.  93 133 

69.  Measurement  of  small  angles  without  auxiliary  mirror.    Table  37 134 

70.  Complementary  fringes.     Figs.  94,  95 135 

71.  Equations 138 

72.  Separated  rigid  vertical  system.     Figs.  96,  97 139 

73.  The  displacement  interferometry  of  long  distances 141 

74.  Theory.    Table  38 142 


PREFACE. 

In  the  present  volume  I  have  pursued  the  work  on  the  interferences  of 
reversed  and  non-reversed  spectra,  begun  in  my  last  report  (Carnegie  Inst. 
Wash.  Pub.  No.  249,  1916),  in  a  variety  of  promising  directions,  such  as  the 
original  investigation  suggested.  It  will  be  remembered  that  the  reversal 
(180°)  here  contemplated  takes  place  on  a  transverse  line  of  the  spectrum 
(i.e.,  a  line  parallel  to  the  Fraunhofer  lines),  which  thereby  becomes  a  line 
of  symmetry  for  the  phenomena.  The  apparatus  has  been  extensively  modi- 
fied, so  as  to  admit  of  measurements  relating  to  individual  fringes.  The 
object  of  such  quantitative  work,  however,  is  to  furnish  a  guide  for  the  devel- 
opment of  the  experiments  and  to  corroborate  equations,  not  to  collate 
standard  data.  These  could  hardly  be  satisfactorily  obtained,  moreover, 
unless  the  work  were  done  with  optical  plates  and  mirrors,  whereas  the  work 
in  this  volume  and  the  preceding  has  been  done  with  ordinary  window-plate 
and  usually  with  film  gratings. 

A  large  part  of  Chapter  I  is  devoted  to  the  treatment  of  prismatic  methods, 
developed  with  the  additional  purpose  of  securing  a  greater  intensity  of  light. 
A  very  curious  intermediate  case  between  the  interferences  of  reversed  and 
non-reversed  spectra  is  the  pronounced  interference  of  spectra  from  the  same 
source,  but  of  different  lengths  (dispersion)  between  red  and  violet.  The 
phenomena  of  crossed  rays  find  a  parallel  occurrence  in  the  present  paper, 
in  the  behavior  of  duplicated  fringes,  when  similar  gratings  or  prisms  disperse 
and  subsequently  recombine  a  beam  of  white  light.  A  type  of  fringes  is 
detected  which  depends  merely  on  the  grating  space  and  is  independent  of 
wave-length.  An  interesting  question  as  to  the  limits  of  micrometer  displace- 
ment within  which  fringes  of  any  kind  are  discernible  (observations  which 
were  at  first  supposed  to  be  due  to  the  degree  of  uniformity  of  interfering 
wave-trains)  is  eventually  shown  to  be  a  necessary  result  of  dispersion. 
Finally,  the  direct  interference  of  divergent  rays  obtained  from  polarizing 
media  is  exhibited. 

In  Chapter  II  the  interferences  of  inverted  spectra,  a  subject  merely 
touched  in  the  preceding  volume,  are  given  greater  prominence.  In  this 
case  one  of  the  two  spectra  from  the  same  source  is  inverted  (180°)  relatively 
to  the  other  on  a  longitudinal  axis  (i.e.,  an  axis  normal  to  the  Fraunhofer 
lines),  which  thus  becomes  a  line  of  symmetry.  In  the  development  of  the 
subject,  spectra  half  reversed  and  spectra  both  reversed  and  inverted  are 
treated  successfully.  In  the  latter  case  the  conditions  of  interference  are 
fulfilled  at  but  a  single  point  in  the  whole  area  of  the  spectrum  field;  and 
yet  the  phenomenon  is  pronounced  and  not  very  difficult  to  realize.  The 
limits  of  micrometer  displacement  within  which  interferences  may  be  obtained 
are  again  determined.  At  the  end  of  the  chapter  it  was  thought  useful  to 
collate  available  equations  in  the  treatment  of  phenomena  of  the  present  kind. 

I 


6  PREFACE. 

The  third  and  fourth  chapters  are  incidental  applications  of  the  displace- 
ment interferometer  and  contain  experiments  on  the  expansion  of  metal  tubes 
by  internal  pressure  and  on  a  promising  method  of  measuring  the  refraction 
of  glass,  irrespective  of  form.  To  carry  out  the  experiments  in  the  last  case 
with  requisite  rigor,  optic  plate-glass  apparatus  would  unquestionably  be 
essential.  Nevertheless  the  tentative  data  obtained  are  noteworthy. 

I  have  begun  in  Chapter  V  the  development  of  displacement  interf  erometry 
in  connection  with  the  older  Jamin-Mach  interferometer,  an  instrument 
which  has  certain  peculiar  advantages  and  is  in  a  measure  complementary 
to  the  Michelson  interferometer.  The  work  was  undertaken  in  connection 
with  the  micromeasurement  of  the  difference  of  heights  of  communicating 
columns  of  liquids,  though  the  latter  had  to  be  abandoned  in  consequence 
of  the  excessive  tremor  in  a  laboratory  surrounded  by  active  city  traffic. 
I  shall  hope,  however,  to  carry  out  such  work  elsewhere. 

The  chief  result  of  Chapter  V  is  the  detection  of  the  achromatic  inter- 
ferences, as  I  have  called  them  for  convenience — interferences  which  are 
ultimately  colors  of  thin  plates  seen  at  oblique  incidence;  but  with  the  new 
interferometer,  and  obtained  with  white  light,  they  are  peculiarly  straight 
and  vivid  and  resemble  a  narrow  group  of  sharp  Fresnellian  fringes  with  the 
central  member  nearly  in  black  and  white.  They  are  capable  of  indefinite 
magnification  and  their  displacement  equivalent  is  a  fraction  of  a  mean  wave- 
length per  fringe.  Notwithstanding  their  strength  and  clearness,  they  are 
so  mobile  in  connection  with  micrometric  displacement  that  in  general  it 
would  be  almost  hopeless  to  attempt  to  find  them  but  for  the  fact  that  they 
coincide  in  adjustment  with  the  centered  ellipses  or  hyperbolae  of  the  spectrum 
fringes  of  the  displacement  interferometer. 

The  fine  white  slit-image  which  is  dispersed  to  produce  the  latter  carries 
the  achromatic  fringes  when  the  slit  is  indefinitely  broadened  or  removed. 
Once  found,  moreover,  they  are  not  sensitive  to  small  differences  of  adjust- 
ment if  a  change  of  focal  plane  is  admissible.  The  chapter  shows  a  curious 
method  for  the  measurement  of  vertical  displacements,  possibly  available 
for  the  detection  of  ether  drag,  which,  though  just  insufficient  in  connection 
with  the  spectrum  fringes,  would  be  promising  in  connection  with  the  achro- 
matic fringes.  Finally,  the  chapter  contains  some  repetitions  of  the  old 
experiments  of  Fizeau  on  the  periodic  evanescence  of  fringes  due  to  the  sodium 
lines.  Curiously  enough,  the  achromatic  fringes  also  show  periodic  recurrence 
sometimes,  which  as  yet  remains  unexplained. 

The  peculiar  adaptability  of  the  new  interferometer  to  the  measurement 
of  small  angles,  either  in  a  horizontal  or  a  vertical  plane,  is  developed  in  the 
final  chapter.  The  ratio  of  the  angular  displacement  of  fringes  to  the  angle 
to  be  measured  (*.*.,  the  rotation  either  of  the  paired  mirrors  or  of  an  auxiliary 
mirror  in  the  apparatus)  may  be  made  enormously  large,  and  the  paper  shows 
cases  in  which,  with  strong  luminous  fringes,  the  angle  to  be  measured  is  mag- 
nified 500  times.  Moreover,  this  is  by  no  means  a  limiting  performance.  Again, 
while  angles  as  small  as  a  few  tenths  of  a  second  or  less  are  measured,  angles 


PREFACE.  7 

as  large  as  several  degrees  come  naturally  within  the  scope  of  the  method. 
Similar  remarks  may  be  made  with  respect  to  the  ratio  of  angular  displace- 
ment and  micrometer  displacement.  Given,  therefore,  an  apparatus  which 
measures  very  small  angles  without  constraint  or  forced  approximations,  the 
measurement  of  long  distances  is  the  next  result  in  order;  for  it  is  merely 
necessary  to  place  the  angle  to  be  measured  at  the  apex  of  the  distance  triangle 
on  the  length  of  the  ray  parallelogram  as  a  base.  This  may  be  done  in  a 
variety  of  ways,  some  of  which  are  shown  in  the  chapter.  The  sensitiveness 
may  again  be  made  remarkably  large. 

The  fringes  here  in  question  are  preferably  the  very  luminous  achromatic 
fringes.  They  have  been  identified  as  ultimately  colors  of  thin  plates,  but 
they  look  like  Fresnel's  fringes.  In  connection  with  this  work,  however, 
another  type  of  fringes  was  detected  obtainable  with  a  fine  slit,  white  light, 
and  in  case  of  centered  spectrum  fringes  when  the  ocular  of  the  telescope 
(or  the  eye)  is  out  of  focus.  These  are  actually  Fresnellian  interferences, 
but  being  made  up  of  broad  concentric  hyperbolic  areas,  brilliantly  comple- 
mentary in  color,  they  resemble  the  lemniscates  of  biaxial  crystals  without 
the  shadows. 

My  thanks  are  due  Miss  Lena  F.  Uhlig,  who  has  assisted  me  efficiently  in 
the  preparation  of  this  volume  for  the  press. 

CARL  BARUS. 
BROWN  UNIVERSITY, 
Providence,  Rhode  Island,  June  1917. 


CHAPTER  I. 

METHODS  FOR  REVERSED  AND  NON-REVERSED  SPECTRUM 
INTERFEROMETRY. 

1.  Introductory. — Thus  far  it  has  been  impossible  to  use  the  fringes  of 
reversed  spectra  individually,  because  of  the  tremor  of  the  apparatus.    It  is 
therefore  desirable  to  endeavor  to  obviate  this  annoyance  as  far  as  possible, 
and  the  end  would  appear  to  be  most  easily  obtainable  if  the  distances  cor- 
responding to  the  same  path-difference  are  made  smaller.    At  the  same  time 
the  results  for  small  distances  will  be  interesting  for  this  very  reason  in 
contrast  to  the  long-distance  methods. 

Furthermore,  the  development  of  different  methods,  with  a  consideration 
of  the  peculiarities  of  each,  will  constitute  an  essential  contribution  to  the 
theory  of  the  phenomena;  for  from  this  the  degree  of  importance  which  is 
to  be  attached  to  the  original  diffraction  at  the  slit  of  the  collimator  (i.e., 
the  limiting  angle  at  the  slit,  within  which  diffracted  rays  must  lie  to  be 
subsequently  capable  of  interference,  whether  reversed  or  inverted)  will 
appear  in  its  relations  to  the  total  dispersion  of  the  system.  The  slit,  however 
fine,  is  still  a  wave-front  of  finite  breadth. 

2.  Apparatus. — In  the  first  experiment,   the  device  with  two  identical 
reflecting  gratings,  GG',  figure  i,  was  firmly  mounted  on  a  massive  spectrom- 
eter, the  four  mirrors,  m,  n,  M,  N,  being  specially  attached.    White  light 
received  from  the  collimator  L  after  two  dispersions  was  viewed  at  the  tele- 
scope T.    Both  gratings  were  on  a  slide  ss,  enlarged  in  figure  2,  set  in  the 
direction  LT  of  the  previous  figure.    The  carriages,  figure  2,  was  provided 
with  universal  joints  (a  with  a  vertical  axis,  b  and  e  with  horizontal  axes 
normal  to  each  other),  while  the  swiveling  of  the  grating  G  was  controlled 
by  set-screws  at  d,  relative  to  the  axle  at  e. 

Unfortunately,  the  displacement  of  the  mirror  M,  figure  i  (on  a  microm- 
eter), passes  the  corresponding  pencil  across  the  face  of  the  grating  G'  and  thus 
virtually  includes  a  fore-and-aft  motion  of  the  latter.  Thus  the  fringes  pass, 
with  rotation,  from  very  fine,  hair-like  striations,  through  a  horizontal  maxi- 
mum of  coarseness,  back  to  vertical  lines  again,  when  homogeneous  light 
and  a  wide  slit  are  employed.  The  annoyances  due  to  tremor,  however,  were 
not  overcome.  Moreover,  there  is  difficulty  in  obtaining  Fraunhofer  lines 
normal  to  the  longitudinal  axis  of  the  spectrum.  This  method  was  therefore 
abandoned. 

The  design  shown  in  figure  3,  with  a  transmitting  grating  at  G  (grating 
space  D  =  3S2Xio-*  cm.)  and  a  stronger  reflecting  grating  at  G'  (D  =  20oX  icr* 
cm.) ,  was  next  tested,  M  being  the  micrometer  mirror.  The  mean  distance  of 
M  from  N  was  about  15  cm.,  from  MN  to  G'  about  10  cm.  and  to  G  40  cm. 

9 


10 


THE   INTERFEROMETRY  OF 


Later  these  distances  were  enlarged.  First-order  spectra  were  used  and  the 
fringes  obtained  easily  and  brilliantly,  particularly  with  mercury  light,  in 
both  green  and  yellow.  They  rotated  as  above,  admitted  a  displacement 
M  of  about  i  cm.  But  they  were  still  too  mobile  to  be  used  individually. 

The  same  design,  figure  3,  was  now  mounted  on  a  round,  heavy  block  of 
cast  iron  B,  30  cm.  in  diameter  and  4  cm.  thick,  the  distance  G  to  MN  being 
about  20  cm.  A  number  of  screw-sockets,  b,  b,  were  drilled  into  B  on  the 
right  and  left,  for  mounting  subsidiary  apparatus.  G'  as  before  was  on  the 
universal  slide  (fig.  2),  movable  in  the  direction  LT.  The  tablets  t,  t',  etc., 
of  G,  M,  N,  and  G'  were  mounted  tentatively  on  standards  of  gas-pipe  1.5  cm. 
in  external  diameter  and  6  cm.  long.  Slight  pressure  by  the  finger-tips  showed 
a  passage  of  several  fringes  across  the  field,  but  the  fringes  were  stationary 
in  the  absence  of  manual  interferences  and  in  spite  of  all  laboratory  tremors. 
A  parallel  arm  of  the  same  pipe  was  therefore  firmly  attached  to  the  stem  of 


3D 


X) 


N  and  M,  each  arm  terminating  in  a  fine  horizontal  set-screw,  s,  s,  below, 
adapted  to  push  against  the  rim  of  the  iron  block.  In  this  way  adequately 
stationary  conditions  and  an  elastic  fine  adjustment  for  superposed  longi- 
tudinal spectrum  axes  were  both  secured  with  advantage.  Similar  elastic 
adjustments  have  been  recently  applied.  It  was  now  possible  to  manipulate 
the  micrometer  at  M  by  hand;  but  a  glass-plate  compensator  C,  rotated  by  a 
tangent  screw  over  a  graduated  arc,  was  also  convenient.  Later  other  types 
were  attached,  including  an  air-compensator,  in  which  path-difference  was 
secured  by  exhausting  the  air  within  a  closed  pipe  provided  with  glass-plate 
ends.  These  contrivances  were  eventually  superfluous,  however,  as  it  was 
found  that  on  reducing  the  rotation  of  the  micrometer-screw  the  latter  could 
be  used  at  once. 

In  case  of  homogeneous  light  and  a  wide  slit,  fringes  were  visible  in  an  ordi- 
nary telescope  for  a  play  of  over  2  cm.  of  the  micrometer-screw,  passing,  how- 
ever, between  extremes  of  fineness.  The  slit  images  are  not  of  equal  breadth, 
if  first-  and  second-order  spectra  are  superposed;  but  if  the  longitudinal  axes 
are  coincident,  any  position  of  the  narrow  image  within  the  broader  produces 
a  wide  vertical  distribution  of  fringes,  usually  more  or  less  horizontal.  They 
are  very  easily  found.  The  sodium  flame  is  too  feeble  for  use.  The  mercury 


REVERSED   AND   NON-REVERSED   SPECTRA. 


11 


arc  is  unfortunately  too  flickering,  so  that  the  fringes  jump  about  and  are 
useless  for  measurement.  Excellently  sharp  quiet  fringes  are  obtained  with 
sunlight  (white),  in  which  the  cross-hatched  interference  pattern  is  nearly 
linear  at  the  line  of  symmetry  of  the  reversed  spectra.  The  fringes  climb 
very  decisively  up  and  down  this  line  with  the  motion  of  the  micrometer, 
reduced  as  suggested.  The  electric  arc  and  a  Nernst  filament  are  equally 
available  as  a  source  of  light.  Finally,  by  suitably  rotating  the  grating  G', 
figure  2,  on  the  axis  e,  by  aid  of  the  set  of  screws  d,  fringes  whose  distance 
apart  is  over  one-third  of  the  width  of  the  telescope  field  may  be  obtained 
quite  sharply.  As  this  distance  represents  but  30 X  io~°  cm.,  there  is  no  diffi- 
culty of  realizing  icr*  cm.  in  case  of  these  long  fringes. 

3.  Measurements.  First-  and  second-order  spectra. — The  steadiness  of  the 
fringes,  even  in  an  agitated  location,  induced  me  to  make  a  few  measurements 
for  orientation.  Accordingly,  the  Fraunhofer  micrometer,  reading  to  icr4  cm., 
was  provided  at  its  screw-head  with  a  light  wooden  wheel  w,  figure  4,  about 


10  cm.  in  diameter  and  3  mm.  thick.  A  groove  was  cut  in  the  circumference 
of  the  wheel,  so  that  a  silk  thread  t  could  be  wrapped  around  it.  The  other 
end  of  the  thread  was  wound  around  a  brass  screw  s,  about  6  mm.  in  diameter, 
turning  in  a  nut,  preferably  of  fiber,  which  was  fastened  to  the  edge  of  the 
table  by  a  small  brass  clamp.  In  this  way  it  was  possible  to  control  the  motion 
of  individual  fringes  crossing  a  fiducial  line  in  the  field  of  the  telescope.  This 
simple  device  worked  surprisingly  well,  a  smoothly  running  micrometer  being 
presupposed.  In  fact,  it  was  possible  to  set  a  fringe  to  a  few  millionths  of  a 
centimeter.  Later  the  micrometer-head  was  grooved  and  a  finer  turning- 
screw  suitably  attached  to  the  base  B  of  the  apparatus.  (Cf.  §  70,  below.) 

The  fringes  should  be  widened  as  far  as  convenient,  by  rotating  the  grating 
on  the  axle  e,  figure  2,  by  aid  of  the  set-screws  d.  In  this  case  they  climb  up 
or  down  the  transverse  strip  as  5  in  figure  4  is  slowly  rotated.  Fringes  moving 
horizontally  are  not  serviceable,  because  they  are  too  near  together.  It  is 
not  difficult  to  obtain  the  single  vertical  line,  black  or  bright,  on  suitable 
rotation  about  e.  On  either  side  of  this  transitional  adjustment  the  fringes 
move  vertically  (climb  or  fall)  in  opposite  directions  for  the  same  micrometer 


12 


THE   INTERFEROMETRY   OF 


displacement.  The  arrow-shaped  forms  are  also  often  satisfactory,  and  may 
be  obtained  by  adjusting  the  two  bright  patches  on  the  reflecting  grating 
into  coincidence,  by  the  eye,  in  the  absence  of  the  telescope.  The  grating  G' 
is  moved  fore  and  aft  for  this  purpose  on  the  slide  s,  figure  2,  until  the  two 
bright  strips  become  one. 

In  making  the  first  adjustment,  I  incidentally  combined  the  first-order 
spectrum  from  N  with  the  second-order  spectrum  from  M,  as  shown  in  figure  5, 
under  the  impression  that  the  wider  D  groups  from  the  latter  were  due  to 
slight  curvatures  of  mirrors.  The  fringes  were  nevertheless  easily  found  and 
showed  no  anomalies,  except  that  observation  had  to  be  made  near  M  or  N. 

It  appears  from  figure  5  that  the  equations  for  this  case  imply 

sin  i  +  sin  0"2  =  2\/A 
sin  *  —  sin  0'2  =  X/A 

and  sin 


where  the  angles  i  and  6  are  equal  (0'2=  0*2).     Thus  sin  t  = 
0=X/2A,  A  being  the  grating  constant  (A  =  2ooXio~6  cm.). 
Hence 

sin  i  =  0.4420  sin  02  =  0.1423 
*  =  26°i4'         02  =  8°n' 

while  from  the  first  grating,  A  =  352X10-*  cm.,  0i  =  9°38';  whence 
<7=i-j-e1  =  3S°52/  8=i—  0=i6°36/ 


--1  =—    =i3 

The  trial  readings  given  in  table  i  of  the  micrometer  for  a  passage  of 
fringes  each  were  found  without  special  precautions,  which  is  equivalent 


TABLE  i. 


Scale  parts. 

io-»  cm. 

19.7 

21.0 
22.2 

23-4 
24.6 
25-8 
27.0 
28.2 

9-85 
10.5 
II.  I 
11.7 
12.3 
12.9 

13-5 
14.1 

to  an  average  of  io-6X3o.i  cm.  per  fringe.  As  the  line  of  symmetry  lay 
very  near  the  two  AA  doublets,  this  is  obviously  an  approach  to  half  a  wave- 
length. For  accurate  work  AA  and  D\D'2  should  be  superposed,  in  which 
case  the  fringes  would  lie  between  and  actually  correspond  to  their  mean 
wave-length. 

A  number  of  measurements  like  the  above  were  now  made  with  different 
types  of  fringes.  The  mean  values  successively  taken  from  3  or  4  batches 
of  20  fringes  each  were 

10^  =  29.4,  30.0,  31.2,  30.6,  29.7,  30.1,  30.0,  30.0,  30.8  cm. 
The  results  were  less  decided  when  long  fringes  were  used.     The  mean 
value  of  the  I0  sets  is  thus  5^Xio6  =  3o.i9  cm.  per  fringe.    Actual  or  approx- 
mate  coincidence  of  the  D  lines  made  no  appreciable  difference. 


REVERSED  AND   NON-REVERSED   SPECTRA. 


13 


In  the  following  results  the  reflection  from  the  mirror  M,  figure  5,  was 
used  in  the  first  order  and  from  N  in  the  second  order,  after  leaving  G' .  Obser- 
vations were  made  near  N,  figure  5.  The  displacement  corresponding  to 
80  fringes  was  successively  0.0024,  0.0024,  0.0024,  0.0024  cm.,  so  that  the 
mean  value 

io6 50  =  30.0  cm. 
agrees  with  the  above. 

Similar  trial  observations  (combined  first  order  from  N  and  second  order 
from  M)  were  made  with  red  light  near  the  C  line  in  series  of  6  with  a  mean 
value  5eXio6  =  34.o  cm.  Again,  near  the  b  line  (green),  giving  5eXio6  =  27.2 
and  27.8  cm.  per  fringe.  These  should  therefore  be  distributed  in  terms  of 
wave-length,  as  in  table  2. 

TABLE  2. 


lo'A 

Ratio. 

io«6e 

Ratio. 

C 
D 
b 

65.6  cm. 
Si-7 

I.  II 
I. 
0.88 

34.0  cm. 
30.2 
27-5 

1.13 
0.91 

and  they  are  as  nearly  as  may  be  expected  in  the  ratio  in  question,  seeing 
that  the  total  displacement  for  60  fringes  does  not  exceed  0.004  cm.  For 
accurate  data  it  would  be  necessary  to  count  many  hundreds  of  fringes,  and 
to  correct  the  8e  values  by  multiplying  by  sec  (02—  0i)/2.  I  have  not  done 
this,  as  the  red  and  green  fringes  are  not  so  distinctly  seen  as  the  yellow. 

4.  Continued.  First=order  spectra.  —  The  apparatus  was  now  readjusted  in 
such  a  way  that  first-order  spectra  were  available  from  both  mirrors.  This 
puts  the  grating  G'  at  a  greater  distance  from  the  line  M  and  N  than  before, 
for  the  angle  02  is  smaller.  A  series  of  trial  results  were  investigated  in  the 
same  manner  as  above,  the  mean  values  from  4  successive  pairs  of  80  fringes 
each  being,  in  three  repetitions, 

5eXio6  =  2g.o,  29.4,  29.5  cm. 
and  from  5  pairs  of  100  fringes  each, 


This  makes  an  average  of 

50Xio'  =  29.25  cm. 

somewhat  smaller  than  half  a  wave-length  of  the  D  light  used.  Unfortu- 
nately, the  screw  at  5  (fig.  4)  here  worked  jerkily,  to  which  the  low  value  is 
probably  due. 

In  this  case  sin  61=\/Di,  where  £>1  =  352Xio-«  cm.  or  0'i  =  9°38';  sin  0'2  = 
X/A,  where  A  =  200X10-*  cm.  or  0'2=i7°9'>  whence 
<r  =  26°47'     and     3  =  7°3i' 

In  a  later  series  of  experiments,  the  play  of  the  screw  s  was  improved,  so 


14  THE   INTERFEROMETRY   OF 

that  it  ran  more  smoothly.    The  following  values  were  found  in  two  repeti- 
tions, from  4  pairs  of  80  fringes  each : 

5eXio6  =  30.i,  30.0  cm. 
and  in  5  pairs  of  100  fringes  each, 

5eXio6  =  29.5  cm. 

If  the  mean  value  of  these  data  is  compounded  with  the  above  mean,  the 
average  is 

5eXio6  =  29.56  cm. 

5.  Continued.  Second=order  spectra. — The  same  phenomenon  was  not 
sought  in  the  two  second-order  spectra  from  G'.  Magnificent  arrows  were 
obtained,  useful  throughout  about  5  mm.  of  the  micrometer-screw,  after  which 
they  lost  clearness.  This  limited  range  could  no  doubt  be  immensely  increased 
if  optical  plate  glass  were  employed  in  place  of  the  ordinary  plate  used.  The 
data  for  pairs  of  observations,  including  60  or  80  fringes,  were  (5  repetitions) 
50X10-*  =  30.3,  30.5,  30.0,  30.8,  31.1  cm.  per  fringe,  giving  a  mean  value  of 
5eXio6=3o.5  cm.  In  the  last  two  measurements  the  sodium  doublets 
coincided. 

In  this  case  sin  6"z  =  aX /Dz,  where  D%  =  2 oo  X  i  o"6  cm.  and  D\  =  3  5  2  X  i cr6  cm. 
(first  grating).  Thus 

cr  =  4S0  44',  5  =  26°28' 

If  the  above  mean  data  are  summarized  the  results  appear  as  follows  (X  = 
58. 93X10-*  cm.): 

G  first  order,  Gf  first  order,  mean  5eX  io6  =  29.56  cm. 
G  first  order,  G'  first  and  second  order,  mean  8eX  io6  =  3o.2  cm. 
G  first  order,  G'  second  order,  mean  50Xio6  =  3o.5  cm. 
And  if  computed  as  5e=X/2  cos  5/2,  these  become 

TABLE  3. 


S*XIO« 

Diff. 

29-53  cm. 
29.78 
30.27 

+0.03  cm. 
+  .42 
+  -23 

The  maximum  error  of  4X10-*  cm.  is  equivalent  to  but  a  little  over  i  per  cent 
of  the  distance  between  fringes,  and  it  would  be  idle  to  suppose  that  the  appa- 
ratus, figure  4,  could  be  set  more  accurately.  In  fact,  the  largest  error  occurs 
in  the  second  set,  which  were  first  made  and  in  which  the  play  of  the  apparatus 
was  inadequately  smooth. 

6.  Theory.— Hence  the  theory  of  the  apparatus  (fig.  6)  may  be  regarded 
as  justified.  Here  the  rays  Y  and  Yf  come  from  the  first  grating  (G  transmit- 
ting), and  after  reflection  from  the  opaque  mirrors  M  and  N  (the  former  on 
a  micrometer)  impinge  on  the  second  reflecting  grating  G',  with  a  smaller 


REVERSED  AND   NON-REVERSED    SPECTRA. 


15 


grating  space,  and  thereafter  interfere  along  the  line  T,  entering  the  telescope. 
To  treat  the  case  the  mirrors  M,  etc.,  may  be  rotated  on  the  axis  T  normal  to 
G'  in  the  position  M\.  G'n  and  G'm  show  the  reflections  of  G'  in  the  mirrors 


N  and  Mi.  We  thus  have  a  case  resembling  the  interferences  of  thin  plates, 
and  if  em  is  the  normal  distance  apart  of  the  mirrors  MI  and  N,  the  displace- 
ment Aem  per  fringe  is  given  by 

\  =  2&em  cos  8/2 

where  6  is  the  angle  between  the  rays  incident  and  reflected  at  the  mirrors. 
This  is  the  equation  used  above.  If  the  mirrors  and  the  reflections  of  the 
gratings  G'  make  angles  <r/2  and  a  with  G',  the  actual  lengths  of  the  rays  (pro- 
longed) before  meeting  to  interfere  terminate  in  e  and  f  respectively.  Let  the 
image  of  G'  be  at  a  normal  distance  e  apart.  Then  e  =  2em  cos  <r/2 ,  for  the  figure 
/#&£  is  a  parallelogram.  If  the  distance  eg  is  called  C  we  may  also  write 

X=ecos  02+Csin  02 
since  C  =  2em  sin  0/2  and  the  angle  of  diffraction  02  =  ((r+$)/2. 

7.  Compensator  measurements.  Sharp  wedge. — With  the  object  of  testing 
the  interferometer  under  a  variety  of  conditions,  measurements  were  made 
with  a  number  of  different  compensators  and  the  experience  obtained  may 
be  briefly  given  here.  The  first  of  these  was  a  very  sharp  wedge,  such  as 
may  be  obtained  from  ordinary  plate  glass.  The  piece  selected,  cut  from  an 
old  mirror,  on  being  calipered,  showed  the  following  dimensions:  Length, 
5  cm.;  thickness  at  ends,  0.375  and  0.367  cm.  Hence  the  angle  of  the  wedge 
is  a  =  0.0016  radian,  or  about  o.  i°.  No  difficulty  is  experienced  from  the  devia- 
tion of  the  rays  for  so  small  an  angle,  though  sometimes  the  fringes  are  unequal 
and  the  lines  presumably  curved.  This  wedge  was  attached  to  a  Fraunhofer 
micrometer  moving  horizontally,  and  the  normality  of  the  rays  passing  through 


16  THE   INTERFEROMETRY  OF 

the  glass  was  found  by  rotating  it  around  an  axis  perpendicular  to  the  rays 
until  the  direction  of  motion  of  the  fringes  was  reversed.  In  view  of  the  small 
angle  a  and  the  micrometric  displacement,  it  was  easy  to  count  single  fringes, 
or  fractions  as  far  as  about  1/30  of  a  fringe,  even  though  the  beam  traversed 
the  glass  twice.  In  the  first  experiment  the  following  data  of  the  horizontal 
displacement,  r,  of  the  wedge  were  found  for  successions  of  7  fringes : 

0.2044  cm.        0.2042        0.2043        0.2067 
0.2058  cm.        0.1976        0.1946        0.18888 

Mean,  0.2008  cm.    Per  fringe,  5r  =  0.02 8 7  cm. 

The  last  three  results  are  low,  the  discrepancies  probably  resulting  from 
slight  wabbling  of  the  micrometer  slide.  In  another  series  made  with  care 
as  to  the  normal  adjustment,  the  horizontal  displacement,  r,  of  the  wedge 
for  successions  of  1 1  parallel  fringes  was 

0.3022  cm.        0.2997        0.2974        0.3002        0.2991        0.3078 
0.3081  cm.        0.3101        0.3170        0.3183        0.3155 

Mean,  0.3014  cm.    Per  fringe,  6r  =  o.o274  cm. 

If  x  be  the  distance  from  apex  of  the  wedge,  its  thickness  is  e=xa,  or  per 
fringe  8e  =  a8x.  The  index  of  refraction  was  found  to  be  /*=  1.526  by  total 
reflection.  Thus,  without  correcting  for  dispersion, 

2(M-i)te  =  X 
and  with  the  above  values 

io-5X5-893  ,. 

a  =  —  — —  =0.0020  radian 

2X0.526X0.028 

This  is  larger  than  the  calipered  value  because  the  rays  go  through  the  wedge 
twice  obliquely.  The  reduction,  however,  would  be  too  complicated  here 
and  will  be  treated  later. 

The  irregularities  above  are  referable  to  the  micrometer,  which  was  not  very 
accurate,  and  no  particular  care  was  taken  with  details.  The  method  is  inter- 
esting as  allowing  of  the  complete  control  of  a  single  fringe;  i.e.,  the  equivalent 
of  30 X  iQ-6  cm.  As  this  corresponds  to  0.028  cm.  on  the  micrometer,  the  dis- 
placement &£  =  o.ooi  is  equivalent  to  io~*  cm.  Furthermore,  the  method  pre- 
sents an  expeditious  means  of  rinding  a=X/2(/i— 1)5#  when  a  is  very  small. 

8.  Continuation.  Revolving  plate.— In  the  next  place,  the  revolving  com- 
pensator C,  figure  3,  was  employed.  This  also  proved  to  be  an  admirable 
device  for  controlling  the  fringes,  and  it  was  much  more  rapid  than  the  pre- 
ceding. Unfortunately  the  computation  is  inconvenient,  as  the  normal 
position  can  not  be  ascertained  with  sufficient  accuracy.  To  find  it,  the  plate 
was  revolved  until  the  fringes  changed  their  direction  of  motion.  This  is  an 
indication  of  the  insertion  of  the  minimum  thickness  of  glass,  but  is  not  sharp 
enough  for  precision.  Hence  data  in  A,  table  4,  are  not  coincident,  i  denoting 
the  angle  of  incidence.  Another  somewhat  better  and  thicker  plate  was  now 
inserted  with  the  results  shown  in  B. 


REVERSED  AND   NON-REVERSED   SPECTRA. 


17 


TABLE  4. 


A.  Same  plate  as  in  the  preceding  work, 
6=0.370  cm. 

B.  Thickness  6=0489  cm. 

No.  of 
fringes. 

j 

i  (probable 
value)  . 

No.  of 
fringes. 

i 

i  (probable 
value)  . 

0 
10 
20 
30 
40 
50 

0°        0°           0° 

44    5-8      7-3 
7.4    8.0      9,5 
9^6    9.7     1  1.  1 
11.4             12.6 
13.0             13.8 

0° 

» 

9-5 

II.O 

12.3 

o 

10 
20 
30 
40 

0°          0°       0° 

6.0      5.5    4.9 
7-9      7-3    7-0 
9-5      8.9    8.5 
10.9    10.3    9.9 

0° 

ti 

8* 

9.6 

The  second  series  here  is  practically  the  mean  of  the 
two,  though  the  reason  for  these  large  discrepancies  is 
not  clear  to  me,  even  in  consideration  of  the  wedge- 
shaped  plates.  The  mean  of  the  results  may,  however, 
be  used  for  computation. 

The  path-increment  introduced  by  the  glass  of  thick- 
ness e  =  0.489  cm.  and  index  of  refraction  /*  =  1.526,  at 
an  angle  of  incidence  i  and  refraction  r  for  n  fringes, 
beginning  at  i  =  o,  may  be  written  (see  fig.  7,  where  J  is 
the  incident  ray) 

cos  (i  -r)        \ 
i  ) 
/ 


cosr 


cosr 


This  is  a  cumbersome  equation.    If  the  angles  i  are  small,  the  cosines  may 
be  expanded  and  then  approximately 

2nX=*((/x-i)r2-K;-r)2) 
which,  since  i=nr  nearly,  may  be  further  simplified  to 


Thus  for  the  second  set  (mean) 

r  =  3.6°        4-8° 
ioBX  =  6.7          6.4 


5-9 
6.9 


6.8° 
6.5  cm. 


The  wave-length  thus  comes  out  very  much  too  large,  but  in  consideration 
of  the  inadequacy  of  the  fiducial  position,  t  =  o°,  this  is  not  unexpected. 
Thus  the  probable  values  of  i  in  the  tables  (computed  from  X  correct)  agree 
with  the  third  series.  In  addition  to  this  the  effect  of  slightly  wedge-shaped 
plates,  etc.,  can  not  be  ignored.  For  the  first  set  (mean  values)  the  results 
are  similar,  being 


7.0 


6.9 


7.0 


7.0  cm. 


if  computed  by  the  approximate  equation.  This  is  again  too  large,  but  the 
probable  value  of  i  computed  from  the  correct  X,  as  before,  agrees  nearly 
with  the  first  series  of  this  set. 


18 


THE   INTERFEROMETRY  OF 


9.  Continuation.    Air  column.— An   air  compensator  was  now   installed 

consisting  of  a  tube  0=15  cm.  long  and  about  2  cm.  in  diameter,  closed  with 
glass  plates.  The  fringes  were  easily  found  and  sharp.  Unfortunately  the 
pump  was  not  quite  tight,  so  that,  on  breaking  the  count  of  fringes  at  low  pres- 
sures, it  was  difficult  to  state  when  the  conditions  had  become  isothermal. 
Hence  the  results  in  table  5  are  rough.  Temp.  19.7°. 

TABLE  5. 


No.  of 
fringes. 

Exhausted 
to  (p). 

dp 
dn 

XIO« 

0 

75.1  cm. 

.... 

£ 

41.8 
o 

n.  I 
n.  I 

59-7  cm. 
60.0 

0 

75-i 

30 

43  <o 

107 

57-6 

70 

0 

10.6 

57-1 

0 

75-i 

.... 

40 

32.1 

10.7 

57-8 

68 

0 

II.2 

60.0 

The  mean  value  thus  appears  as  X  = 

tions  used  are 

(i)  »X 


for  sodium  light.    The  equa- 


where n  is  the  number  of  fringes  counted,  e  the  tube-length,  and 

of  refraction  of  air.   Again, 

(2)  £=CV-i)t? 


the  index 


where  p  is  the  pressure,  tf  the  absolute  temperature,  and  the  constant  C 
computed  from  normal  conditions  (76  cm.  and  o°  C.)  is  (Mascart's  values) 
(7  =  952.6.  Hence 

(3)  e  *        e    d  e  d 


Cnd 


C&dn 


when  #  is  constant.  It  is  this  assumption  which  is  not  quite  guaranteed  above. 
To  obviate  this  in  the  following  experiments,  the  total  number  of  fringes 
were  counted  (table  6)  from  exhaustion  to  plenum.  Their  number  was  definite 
to  the  fraction  of  a  fringe. 

TABLE  6. 


Temp.  19.3°  C.;  C  =  i$  cm. 

No.  of 
fringes. 

Exhausted 
to(£). 

dp 
dn 

Aio8 

o 

69.5 
o 

69-5 
o 

69.5 

75-8  cm. 

0 

75-8 

0 

75-8 

0 

1.090 
1.090 
1.090 

58.8  cm. 
58.8 
58.8 

REVERSED   AND   NON-REVERSED   SPECTRA.  19 

These  results  are  correct  to  0.5  per  cent  and  are  as  close  as  the  estimation  of 
p,  c,  &,  and  fractions  of  a  fringe  will  warrant.  If  results  of  precision  were 
aimed  at,  a  long  tube  should  of  course  be  used.  What  was  particularly 
marked  in  these  experiments  was  the  motion  of  fringes  in  the  passage  from 
any  approximately  adiabatic  to  isothermal  conditions  and  on  approaching  a 
plenum  of  air. 

Since  the  refraction  depends  on  density,  there  should  not  (apparently)  be 
any  motion  at  all;  but  the  thin  tube  is  always  more  nearly  isothermal  than 
the  much  larger  barrel  of  the  air-pump.  As  a  consequence  there  is  residual 
expansion  from  the  former  to  the  latter. 

10.  Continuation.  Babinet  compensator. — The  behavior  of  an  old  Babinet 
compensator,  placed  nearly  normal  to  one  of  the  beams,  figure  3,  was  peculiar, 
though  the  fringes  were  clear  and  easily  controlled.  The  dimensions  of  the 
right-handed  quartz  wedge  were  roughly  calipered  and  found  to  be:  length, 
4.2  cm.;  thickness  at  ends,  1.017  and  0.934  cm.  Thus  there  is  a  grade  of 
0.083/4.2  =0.0193,  or  something  over  i°  of  arc.  A  vertical  displacement  of 
2.5  cm.  of  this  wedge  was  available  behind  the  stationary  counteracting 
left-handed  wedge.  / 

The  fringes  were  not  uniform  and  they  required  an  inclination  ^ 

to  the  vertical  of  the  rulings  of  the  grating  G'.    The  fringes  were  /i 

evidently  curved  lines,  intersected  by  the  vertical  strip  within  A^J 

which  they  are  visible.    Consequently  they  appeared  as  in  figure  I  /} 

8,  with  linear  elements  in  the  middle,  shortening  into  dots  at  either  / 

end  of  the  strip.    On  motion  of  the  compensator  wedge  they 
moved  toward  or  from  the  center  of  symmetry,  as  is  also  indicated    g 
in  the  figure.    Tiled  forms  were  frequent.    The  most  interesting 
feature,  however,  was  their  alternate  appearance  and  evanescence 
in  cycles.    While  the  wedge  was  moved  over  2.5  cm.  of  its  length,         J  \ 
7  of  these  cycles  appeared  and  vanished,  each  consisting  of  about        TV? 
36  to  40  fringes.     The  disappearance  was  not  always  quite  com-  \  I 

plete,  but  the  fringes  could  not  be  restored  by  any  adjustment  for  ^ 

coincidence  of  spectra. 

An  attempt  was  made  to  find  the  angle  of  the  quartz  wedge  by  the  first 
method.  Data,  0.0023,  0.0024,  0.0024  cm.,  were  found  for  the  displacement 
of  the  micrometer  per  fringe.  Hence,  apart  from  dispersion, 

io~5Xs.893  ,. 

a=  —  = =  0.022  radian 

2X0.5442X0.0024 

which,  as  in  case  of  the  glass  plate,  is  again  slightly  above  the  calipered  value. 

In  another  somewhat  thinner  Babinet  compensator  the  constants  were: 
length,  3.35  cm.;  thickness,  small  end,  0.494  cm.,  large  end,  0.496  cm.  The 
prism  angle  is  (2  =  0.062/3  35  =  0.0185  radian,  also  about  i°. 

In  this  case  there  was  no  periodic  phenomenon,  but  in  its  place  the  degree 
of  longitudinal  coincidence  of  the  axes  of  the  two  spectra  continually  changed. 


20 


THE   INTERFEROMETRY   OF 


The  fringes  at  once  sharpened,  however,  on  readjustment  of  either  mirror, 
indicating  a  continuous  small  change  of  deviation,  due  to  curvature,  probably, 
in  the  quartz  wedge.  In  the  preceding  periodic  case,  no  readjustment  of 
deviation  sufficed  to  restore  the  fringes.  The  wedge  was  now  detached  and 
used  alone.  In  spite  of  the  relatively  large  angle  (i°),  no  difficulty  was  ex- 
perienced in  adjusting  or  controlling  the  fringes;  but  the  face  curvature  just 
suggested  appeared  as  before,  so  that  readjustment  for  varying  wedge-angle 
was  required  from  time  to  time. 

11.  Micrometer  displacement  of  the  second  grating,  G'  (fig.  3). — In  the 
preceding  report  (Carnegie  Inst.  Wash.  Publication  249,  Chapter  III,  §  28)  it 
was  shown  that  if  the  angle  between  the  gratings  G  and  G'  is  <p  and  the  angle 
between  the  mirrors  M  and  N  (which  in  a  symmetrical  adjustment  would  be 
180°— (0i+  02),  0i  and  02  being  the  angles  of  diffraction  at  G  and  G'  for  normal 
incidence  at  G)  is  decreased  by  a,  so  that  the  adjustment  is  non-symmetrical, 
then  the  displacement  8e  of  the  grating  G'  per  fringe  will  be  very  nearly 

X  cos2  02 
2  (a — <p}  sin  02 

if  a  and  <p  are  small.  Here  a  is  effectively  the  angle  between  the  mirrors  M 
andW,  since,  if  M  is  rotated  180°  on  the  line  of  symmetry  (normal  to  the 
grating  G),  the  two  mirrors  would  intersect  at  an  angle  a.  The  result  of  fore- 
and-aft  motion  thus  depends  on  the  angle  a— <p,  and  if  a  =  <f>,  de=  oo  per 
fringe;  i.e.,  fore-and-aft  motion  would  produce  no  result.  This  is  necessarily 

TABLE  7. 


»'=  increasing;  d  =  decreasing. 

No.  of 

fringes. 

Total 
displace- 
ment. 

Mean  8e 
displace- 
ment per 
fringe. 

*3 

0.0234  cm. 

0.0088  cm. 

3 

267 

3 

293 

»  4 

.0302 

.0080 

4 

317 

4 

3i8 

4 

346 

4 

330 

d4 

.0285 

.0073 

4 

285 

4 

287 



4 

315 

4 

288 

the  case  when  but  a  single  grating  is  used,  as  in  the  earlier  methods.  In  the 
case  of  two  gratings,  however,  it  is  not  only  difficult  to  make  a  perfectly 
symmetrical  adjustment  of  mirrors  and  grating,  but  it  would  not  be  of  any 


REVERSED   AND   NON-REVERSED   SPECTRA. 


21 


special  advantage.  Hence  the  fore-and-aft  displacement  e  of  the  grating  G' 
will  probably  be  accompanied  by  a  slow  motion  of  the  fringes,  from  which 
the  angle  a— <p  may  be  computed. 

The  experiments  recorded  in  table  7  were  made  with  the  grating  G'  on  a 
micrometer-slide  moving  normally  to  the  face  of  the  grating.  With  the 
mirrors,  etc.,  placed  so  that  optical  paths  were  nearly  equal,  the  adjustment 
screws  on  M  and  N  sufficed  to  bring  the  fringes  strongly  into  view.  Succes- 
sions of  3  and  of  4  fringes  were  tested,  as  these  required  an  adequately  large 
displacement  of  the  micrometer,  which  was  moved  both  forward  and  backward. 

The  mean  of  the  three  results  is  de  =  0.008  cm.  per  fringe.  The  data  are 
not  smooth,  because  the  micrometer  placed  between  the  mirrors  M  and  N  is 
in  an  inconvenient  position  for  manipulation.  The  different  sets  of  values, 
moreover,  correspond  to  different  adjustments  and  therefore  to  slightly  differ- 
ent values  of  a — <p .  As  an  order  of  values  only  is  wanted,  it  was  not  considered 
worth  while  to  remedy  the  deficiencies. 

In  accordance  with  the  equation  given,  if  8e  =  0.008  cm.,  X  =  58.9  X  lo"6  cm., 
62=20°  be  inserted, 

X  cos2  02 

a  —  <p  = =0.0095  radian  =  0.54 

2  oe  sin  02 

The  adjustment  is  thus  about  half  a  degree  out  of  symmetry,  a  result 
which  in  case  of  improvised  apparatus  is  inevitable  and  moreover  without 
significance  in  the  precision  of  the  method. 

12.  Prism  method.  Reflection.— The  grating  G  was  now  removed  and 
replaced  by  a  silvered  prism,  as  shown  in  figure  9  (P,  prism;  M,  N,  mirrors; 
G,  grating;  T,  telescope).  A  small  prism  angle,  <f>,  is  essential  (?  =  18°,  about), 
as  a  large  divergence  of  rays  would 
not  be  accommodated  on  the  interfer- 
ometer, figure  3 .  The  fringes  were  found 
without  difficulty,  in  the  second  order, 
the  arc  lamp  being  used.  They  are  also 
easily  distorted,  if  the  edge  of  the  prism 
is  not  parallel  to  the  rulings  of  the  grat- 
ing. In  such  a  case  the  symmetrical 
arrow-shaped  forms  become  one-sided 
and,  as  it  were,  curved  or  faintly 
fringed  beyond  the  limits  of  the  strip. 
To  get  the  best  adjustment,  the  lamp 
should  shed  about  the  same  amount  of  undeviated  light  from  both  faces'  of 
the  prism,  on  a  screen  temporarily  placed  behind  it.  The  illuminated  strips 
on  the  grating  must  coincide  to  the  eye  while  making  the  fore-and-aft  adjust- 
ment. Finally,  the  grating  is  to  be  slowly  rotated  on  the  axis  normal  to 
itself,  until  fringes  of  satisfactory  shape  and  size  appear.  Naturally  this 
is  done  through  the  telescope,  and  a  readjustment  of  the  longitudinal  axes 


22  THE   INTERFEROMETRY  OF 

of  the  spectra  is  necessary  after  each  step  of  rotation.  Fringes  so  obtained 
are  as  good  as  those  obtained  by  any  other  method. 

The  range  within  which  the  fringes  are  sharp  is  small,  not  exceeding  2  mm. 
of  displacement  of  the  micrometer  mirror,  M.  A  partial  reason  for  this  will 
appear  from  figure  9  and  results  from  the  fact  that  the  illumination  on  the 
grating  due  to  M  moves  laterally  across  the  stationary  strip  due  to  N.  Clearly 
if  the  latter  were  also  on  a  micrometer  it  might,  in  turn,  be  displaced  relatively 
to  the  direction  of  M  and  restore  the  fringes  to  full  brilliancy.  The  range 
in  this  case  may  be  increased  till  either  illuminated  strip  gets  beyond  the 
edges  of  the  grating.  This  test  will  presently  be  made. 

If  the  prism  angle  is  <p  and  the  angle  of  diffraction  for  normal  incidence  is  0, 
the  angle  8,  between  the  incident  and  reflected  ray  at  M,  is 


Thus  e  tan  5/2  is  the  displacement  of  the  strip  of  light  on  the  mirror  M, 
if  e  is  the  normal  displacement  of  the  latter.  Hence  the  corresponding  dis- 
placement x  on  the  grating  is 

*  =  2*sin(S/2)/cos0 

If  6  be  the  distance  from  the  prism  to  the  light  spot  reflected  on  M,  and  c 
the  distance  from  there  to  the  bright  spot  on  the  grating,  <p  may  be  com- 
puted as 

_  Csin  B  _  2\c 
b  b 

for  the  spectra  are  in  the  second  order. 
The  data  are: 

io6X  =  58.93  cm.;  6  =  38.0  cm.;  £  =  20.4  cm.;  Z?  =  2ooXio-6  cm. 
Whence 

¥>  =  i8°i2'  0==36°6'            5  =  17°  54';  5/2  =  8°  57' 
Hence 

_  25  X  0.1556 

0.808 

if  e  =  0.2 5  cm.,  as  found.  Thus  the  rays  of  the  same  origin,  or  rays  capable 
of  interfering,  are  found  in  a  vertical  strip  on  the  grating,  not  more  than  i  mm. 
wide.  It  is  interesting  to  note  that  the  fringes  vanish  by  becoming  coarser 
and  wider,  corresponding  to  the  narrowing  of  effective  edges  in  contact. 

The  attempt  to  produce  these  fringes  with  homogeneous  (sodium)  light  and 
a  wide  slit  again  failed,  although  much  time  was  spent  in  the  endeavor.  Even 
with  a  narrow  slit  and  accentuated  sodium  lines  (impregnated  arc)  the  phe- 
nomenon may  be  produced  between  the  doublets,  however  close  together,  but 
it  fails  to  appear  with  the  same  adjustment  when  two  corresponding  lines 
coincide.  I  was  only  able  to  produce  it  in  a  continuous  spectrum,  between 
the  two  doublets  and  with  a  fine  slit.  It  is  very  important  to  ascertain  the 
reason. 


REVERSED  AND   NON-REVERSED   SPECTRA. 


23 


Both  mirrors,  M  and  N,  were  now  placed  on  micrometers  moving  nearly 
normal  to  their  faces.  Beginning  with  a  coincidence  of  the  illuminated  strips 
on  the  grating,  the  M  micrometer  was  moved  until  the  fringes  disappeared. 
The  N  micrometer  was  then  moved  in  the  same  direction,  until  the  reappearing 
fringes  passed  through  an  optimum  and  finally  vanished,  in  turn.  There- 
after the  M  micrometer  was  displaced  again,  always  in  the  given  direction, 
and  the  same  cycle  repeated,  etc.  It  was  possible  to  pass  through  about 
8  cycles  with  each  micrometer  before  the  illumination  reached  the  edge  of 
the  grating,  each  cycle  corresponding  to  a  displacement  of  about  2  mm.  for 
a  single  mirror;  but  a  total  displacement  of  2.5  cm.  was  registered,  which 
would  obviously  have  been  increased  much  further  if  the  grating  had  been 
wider.  The  data  given  in  table  8  give  a  concrete  example : 

TABLE  8. 


Position 

<&N. 

Advance 

otN. 

Remarks. 

Position 
olN. 

Advance 
otN. 

Remarks. 

2.42  cm. 

Broad  arrows 

1.75  cm. 



Vertical  lines 

2.20 

J-54 

0.22  cm. 

M  advanced 

0.21  cm. 

M  advanced 

2.2O 

Narrow  arrows 

1-54 

Vertical  lines 

1.98 

.... 

i-37 

.24 

M  advanced 

1*7 

M  advanced 

I.98 

i-75 

Upright  lines,  in- 
clination changed 

•23 

M  advanced 

As  both  mirrors  move  in  the  same  direction,  the  two  illuminated  strips  on 
the  grating  gradually  separate  until  they  are  quite  distinct.  Meanwhile  the 
fringes  pass  with  rotation  from  the  original  sagittate  forms  to  very  fine  hair- 
like  striations ;  whereas  the  part  of  the  spectrum  within  which  the  former 
occur  is  less  than  the  distance  apart  of  the  sodium  lines  (doublets),  the  hair- 
lines are  visible  within  a  strip  of  spectrum  many  times  as  broad  as  the  sodium 
doublet.  Ten  such  lines  may  be  visible.  In  good  adjustments  the  sagittate 
forms  are  seen  to  be  a  nest  of  very  eccentric,  identical  hyperbolas,  as  in  figure 
10,  A,  arranged  or  strung  on  the  same  major  axis.  The  vertices  a  are  therefore 
thick  and  pronounced,  but  taper  rapidly  down  into  hair-lines,  6,  &',  on  both 
sides.  Frequently  but  half  of  the  coarse  vertices,  a,  abundantly  fringed  on 
one  side,  6  or  V,  appear.  Nevertheless  this  does  not  seem  to  be  an  exhaustive 
description  of  the  phenomena;  for  it  is  not  uncommon,  when  partial  hyper- 
bolas appear,  to  find  the  striations  (which  are  always  faint)  in  the  same 
direction  on  both  sides,  as  in  figure  10,  B\  i.e.,  the  striations  are  apt  to  be 
non-symmetrical  on  the  two  sides,  as  if  they  constituted  a  second  diffraction 
phenomenon  superimposed  on  the  first  phenomenon.  Roof -shaped  forms 
(fig.  10,  Q,  strongly  dotted,  are  also  common,  often  irregularly  awned. 

Figure  1 1  may  be  consulted  to  further  elucidate  the  subjects  under  con- 
sideration. G  is  the  grating,  PP'  the  principal  plane  of  the  objective  of  the 
telescope,  a  and  6  are  two  rays  interfering  at  the  focus  /,  and  leaving  the 
grating  parallel  and  symmetrically  placed  to  the  axial  ray  of.  The  passage 


24 


THE  INTERFEROMETRY  OF 


of  the  coarse  sagittate  phenomena  into  the  hair-like  striations,  as  a  and  b 
move  farther  apart,  may  then  be  accounted  for  in  accordance  with  the  general 
theory  of  diffraction;  i.e.,  if  the  distance  apart  of  a  and  b  is  d  and  the  principal 
focal  distance  of  is  R, 

-  =  — 
d  ~  R 

where  z  is  the  distance  between  the  two  fringes  of  wave-length  X.  Hence  z 
will  increase  as  d  decreases,  agreeing  with  the  effect  of  fore-and-aft  motion, 
or  with  the  effect  of  simultaneous,  large  (2.5  cm.)  displacement  of  both 
mirrors,  neither  of  which  destroys  the  symmetry  of  the  interfering  rays. 


10 


The  motion  of  a  single  mirror,  M  or  N,  for  instance,  does  destroy  the  sym- 
metry, and  it  was  shown  in  §  12  that  the  limiting  range  of  displacement  of 
0.25  cm.  moves  either  a  or  6  0.096  cm.  out  of  symmetry.  The  interferences 
thus  vanish  without  much  changing  in  form  or  size,  and  vanish  in  all  focal 
planes. 

The  breadth  of  the  blades  of  light  aa'  and  bb',  figure  n,  capable  of  inter- 
fering is  x  on  the  grating  and 

x  cos  0  =  0.096X0.808  =  0.078  cm. 

normally.  Since  the  rays  are  parallel  after  leaving  the  collimator,  this  would 
be  about  half  the  breadth  of  the  effective  beam  on  the  objective  of  this  appur- 
tenance. Thus  2X0.0776  =  0.155001.,  increased  by  the  width  of  the  refracting 
edge  of  the  prism,  is  the  width  of  the  strip  of  white  light  which,  after  separa- 
tion by  the  knife-edge  of  the  prism,  furnished  the  two  component  beams  which 
potentially  interfere  on  recombination.  It  is  reasonable  to  suppose  that  the 
elements  of  these  beams  come  from  a  common  source  and  that  the  width  in 
question  is  produced  by  the  diffraction  of  the  slit. 

This  datum  is  more  appropriately  reduced  to  the  angle  at  the  slit  a,  within 
which  lie  the  rays  capable  of  interfering  with  each  other  after  the  interferom- 
eter cleavage.  As  the  collimator  used  was  /  =  22  cm.  from  slit  to  lens, 

a  =  2x  cos  6/1  =  o.i  5 5/2 2  =  0.0070 

Hence  the  angular  width  of  the  wedge  of  white  light,  with  its  apex  at  the  slit 
of  the  collimator  and  containing  all  the  rays  which  can  mutually  interfere, 
is  about  0.007  radian,  or  less  than  half  a  degree  of  arc.  One  would  infer  that 
a  long  (I)  collimator  (i.e.,  one  with  weak  objective)  is  advantageous,  as  the 
blade  of  parallel  rays  issuing  is  proportionally  wide  and  the  range  of  displace- 


REVERSED  AND   NON-REVERSED   SPECTRA.  25 

ment  at  M  or  N  larger.  Similarly,  divergence  subsequently  imparted  by 
dispersion  (prism,  grating),  before  the  rays  reach  the  mirrors,  M,  N,  should 
have  the  same  effect.  The  results  obtained  for  dispersion  bear  this  out,  but 
not  those  for  a  long  collimator.  Moreover,  the  width  of  the  slit,  so  long  as 
the  Fraunhofer  lines  do  not  vanish,  is  of  no  consequence.  It  thus  seems 
tenable  (to  be  carefully  investigated  below)  that  the  positive  effect  of  dis- 
persion has  a  deeper  significance,  bearing  directly  on  the  structure  of  the 
interfering  wave- trains — i.e.,  the  length  of  the  coordinated,  uniform  wave- 
train  is  possibly  greater  as  the  dispersion  to  which  the  wave-train  has  been 
subjected  is  greater.  Two  parts  of  it  will  therefore  fit  over  a  correspondingly 
longer  range  of  path-difference. 

A  number  of  other  results  point  in  the  same  direction.  Thus,  I  may  again 
point  to  the  impossibility  of  obtaining  fringes  with  homogeneous  light  and  a 
wide  slit,  whereas  two  identical  sodium  lines  (D\  and  D'\),  superposed,  show 
the  interferences  strongly.  The  lines  actually  become  helical  in  shape  and 
much  broader.  The  range  of  displacement  of  N  may  be  decreased  from  0.25 
cm.  to  o.io  cm.  by  narrowing  the  beam  emerging  from  the  collimator  with  a 
slotted  screen,  while  the  fringes  themselves  are  coarsened  by  this  process. 
With  the  screen  removed  the  fringes  are  not  only  sharper  and  finer,  but 
apparently  they  may  be  seen  to  slowly  move  laterally  across  the  fiducial 
sodium  lines.  This  is  in  accord  with  the  increased  range  of  displacement  of 
the  mirror.  The  observation,  however,  is  complicated  by  the  fact  that  the 
sodium  doublets  are  not  quite  in  the  same  focal  plane.  The  fringes  must, 
in  a  reduced  case,  lie  midway  between  them,  in  the  line  of  symmetry  of  the 
spectra. 

13.  Prismatic  refraction. — The  method  indicated  in  figure  12  (P,  prism; 
M,N,  mirrors;  G,  grating;  T,  telescope)  was  next  tested  for  small  distances  and 
the  experiments  begun  in  the  third  order  of  spectra  of  the  grating  G.  The 
prism  was  a  small  right-angled  sample,  with  faces  only  about  i  cm.  square; 
but  it  sufficed  very  well.  Its  distance  from  the  grating  being  about  13  cm. 
and  the  illuminated  spots  on  the  mirrors  18.8  cm.  apart,  the  mirrors  were 
nearly  normal  to  each  other.  In  fact,  as  6  in  the  third  order  is  about  62° 
and  i'  about  28°,  5  =  34°  and  a  =  90°.  Hence,  on  displacing  the  micrometer 
mirrors  M  or  N,  the  illuminated  strips  move  relatively  rapidly  across  the  face 
of  the  grating.  Nevertheless,  the  fringes  are  easily  found  and  controlled. 
Their  range  of  visibility  is  larger  than  in  the  cases  of  the  preceding  paragraph. 
They  remain  in  view  for  normal  displacement  of  M  of  3  to  4  mm.,  passing 
from  hair-like  striations,  through  sharp  arrows,  back  to  the  hair-like  forms. 
The  range  has  thus  been  increased  by  the  dispersion.  The  arrows  are  of 
the  type  shown  in  figure  13,  with  reentrant  sides  and  part  of  the  outline 
accentuated. 

In  the  second  order  of  spectra  from  G,  the  phenomena  were  much  the  same, 
but  far  more  brilliant.  The  arrows  were  now  evenly  wedge-shaped  and  very 
slender.  The  fringes  entered  as  nearly  vertical  hair-like  striations,  and,  after 


26 


THE   INTERFEROMETRY  OF 


passing  the  optimum,  vanished  as  inflated  arrows.  The  range  of  visibility 
was,  as  before,  about  3.5  mm.,  so  that  the  change  of  order  has  not  had  any 
further  marked  effect,  such  as  might  be  anticipated.  As  in  the  preceding 
paragraph,  if  the  impinging  collimated  beam  is  narrowed,  the  range  of  visi- 
bility decreases;  in  fact,  the  arrows  themselves  are  reduced  to  slightly  oblique 
lines.  Within  the  limits  given  the  fringes  are  well  adapted  for  interferometry. 

First-order  spectra  are  not  available  because  of  the  large  value  of  i'  in  the 
case  of  the  right-angled  prism. 

Taking  the  results  of  the  last  two  paragraphs  together,  the  increase  of  the 
range  of  displacement  is  due  to  the  dispersion  of  the  prism.  The  breadth  of 
the  pencil,  diffracted  at  the  slit,  after  leaving  the  collimator  and  prism, 


12 


13 


increases.  It  was  shown  in  the  earlier  report  that  inversion  of  spectra  on  a 
longitudinal  axis  does  not  preclude  the  possibility  of  interference.  Taken  as 
a  whole,  therefore,  the  present  results  have  a  direct  bearing  on  Huyghens's 
principle. 

14.  Prism  methods  without  grating. — A  more  interesting  method,  in  some 
respects,  in  which  the  grating  is  entirely  dispensed  with,  is  shown  in  figure 
14.  L  is  the  beam  of  white  light  from  a  collimator,  P  a  refracting  prism  (here 
with  a  60°  prism  angle),  M  and  N  the  opaque  mirrors,  with  either  or  both 
on  a  micrometer,  P'  a  silvered  reflecting  prism  (here  right-angled).  The 
telescope  is  at  T  and  should  have  high  magnification.  The  rays  L  are  refracted 
into  a  b  c  and  a'  b'  c'  and  the  two  spectra  observed  by  the  telescope  at  T. 
Each  of  the  prisms  should  be  on  3  adjustment  screws,  as  well  as  the  mirrors. 
P  must  be  revolvable  slightly  around  a  vertical  axis  and  capable  of  fore-and- 
aft  motion.  P'  is  preferably  a  large  prism  placed  on  a  tablet.  The  rays  b  and 
V  are  made  collinear  before  P'  is  inserted,  and  both  the  rays  c  and  c'  must 
come  from  near  its  edge. 


REVERSED  AND   NON-REVERSED   SPECTRA. 


27 


The  fringes  are  strong  and  large  and  lie  within  a  remarkably  wide  trans- 
verse strip.  This  may  be  10  or  20  times  as  wide  as  the  D\  Dz  doublets,  which, 
in  view  of  the  small  dispersion,  are  hardly  separated.  For  the  same  reason, 
moreover,  the  range  of  displacement  of  M  within  which  fringes  are  visible 
rarely  reaches  0.5  mm.  Within  this  the  fringes  grow  from  the  fine  hair-lines, 
usually  oblique,  to  their  maximum  coarseness.  Apart  from  the  small  range 
of  displacement,  these  fringes  are  available  for  measurement.  If  both  mirrors 
M  and  N  are  on  micrometers,  they  may  be  brought  forward  or  the  reverse, 
alternately,  and  the  range  increased  5  or  10  times. 

To  change  the  form  of  the  fringes,  the  first  prism,  P,  may  be  tilted  slightly 
on  an  axis  parallel  to  LT,  figure  14.  The  fringes  then  pass  through  a  maximum 
in  the  vertical  direction  (linear  phenomenon).  Fore-and-aft  motion  of  P 
rotates  the  fringes,  partially,  toward  the  horizontal;  but,  as  a  rule,  the  com- 


14 


ponent  beams  b  and  bf  pass  beyond  the  edge  of  P'  and  the  fringes  vanish. 
Just  before  this  (the  spectra  separating),  the  strip  within  which  the  fringes 
lie  widens  enormously.  In  other  words,  the  breadth  of  the  phenomenon 
depends  on  diffraction,  not  on  dispersion,  so  that  even  though  the  prism  P 
scarcely  separates  the  V  lines,  the  striated  strip  has  about  the  same  width  as 
when  it  is  produced  by  highly  resolving  gratings. 

It  is  preferable  to  use  sunlight  directly  (without  a  long-focus  condensing 
lens),  as  there  is  a  superabundance  of  light.  The  best  results  are  attained 
with  a  large  collimator.  A  spectacle  lens  with  a  focal  distance  of  i  meter  is 
excellent.  The  range  of  displacement  of  M  is  not  increased,  but  the  spectra 
and  fringes  become  very  sharp.  If,  with  the  large  collimator,  the  spectra 
are  just  separated  in  the  field  of  the  telescope,  by  fore-and-aft  motion  of  P, 
a  magnificent  display  appears,  resembling  a  thick,  twisted  golden  cord.  With 
further  separation  confocal  elliptic  fringes  often  cross  the  gap,  as  in  figure  15. 
Here  a  and  j8  are  graphs  suggesting  the  wave-lengths  of  the  two  spectra,  g 
being  the  gap  or  deficient  overlapping.  The  appearance  in  the  telescope  is 
shown  at  7,  5  and  S'  being  the  spectra.  When  the  fringes  are  erect,  huge 
vertical  furrows  may  lie  in  the  gap.  When  the  gap  is  closed,  the  linear  phe- 
nomenon reappears.  These  enlarged  fringes  vanish,  however,  within  0.25 
mm.  of  displacement  at  M.  On  the  other  hand,  when  the  spectra  are  made 


28  THE   INTERFEROMETRY  OF 

to  overlap  considerably  (by  fore-and-aft  motion  of  P)  the  fringes  become 
fine  and  vertical  and  the  parallel  blades  of  light,  which  interfere  at  the  focus 
of  the  telescope,  are  0.5  to  i  cm.  apart  at  the  objective. 

In  further  experiments,  screens  s,  s'  (fig.  14)  were  placed  in  the  paths  of 
the  pencils  b  b',  so  that  they  were  compelled  to  pass  through  vertical  slits 
0.5  mm.  wide  in  the  screens.  In  this  way  the  interfering  rays  were  identified. 
The  first  vertical  hair-line  fringes  came  from  rays  about  5  mm.  behind  the 
edge  of  the  prism  P'.  Hence  the  pencils  were  here  about  1.2  cm.  apart  when 
they  entered  the  telescope.  The  largest  and  last  of  the  fringes  came  from 
close  to  the  edge  of  P'.  The  experiment  was  varied  as  follows :  Supposing 
both  screens  5  and  s'  placed  as  far  to  the  rear  as  the  visibility  of  fringes  per- 
mits; let  the  former,  s,  be  slowly  pushed  forward.  The  fringes  then  contract 
from  the  very  broad  set,  figure  16,  case  i,  to  the  strong  and  narrow  set  2 
(which  is  a  mere  line  for  a  full  wave-front),  and  then  expand  again  to  case  3. 
If,  now,  s  is  left  in  place  and  s'  moved  forward  slowly  in  the  same  way,  the 
identical  contraction  and  expansion,  cases  i,  2,  3,  are  reproduced.  The  screen 
5'  may  then  be  left  in  place  and  5  in  turn  slowly  moved  forward  with  the  same 
results,  etc.  (there  may  be  6  alternations),  until  finally  the  effective  parts  of 
the  pencils  b  and  b'  are  beyond  the  edge  of  the  prism  P'.  In  case  2  the  two 
slits  s  and  5'  are  obviously  symmetrical  to  the  interfering  rays,  whereas  in 
cases  i  and  3  the  diagonally  opposite  edges  of  the  slits  5  and  s'  limit  the  effi- 
cient pencils  to  a  sheet.  If  the  edge  of  the  prism  were  truly  a  knife-edge,  the 
last  fringes  would  be  very  large,  since  the  distance  cc',  figure  14,  would  vanish. 
If  the  fringes  are  vertical  (obtained  by  tilting  P  around  an  axis  parallel  to 
LT),  the  case  2  is  given  by  2  or  3  strong  vertical  lines,  whereas  i  and  3 
consist  of  10  or  20  lines,  all  of  about  the  same  width  and  distance  apart.  If 
the  slits  5,  s'  are  finer  (i  mm.),  the  fringes  are  throughout  sharper.  A  single 
displacement,  i,  2,  3,  corresponds  to  about  2  mm.  When  the  edge  of  P'  is 
approached  the  case  2  often  shows  vertical  strands  of  fringes,  a  strong  central 
strand,  and  two  or  three  fainter  ones  either  side  of  it.  The  cases  i  and  3  are 
not  stranded. 

A  similar  result  (passage  of  case  2  into  3,  fig.  16)  may  be  produced  by 
moving  P  forward,  the  case  3  appearing  just  before  the  pencils  b  b'  leave  the 
edge  of  P'.  Again,  when  M  is  moved  rearward,  when  both  b  and  b'  are  near 
the  edge  of  P',  the  cases  2  and  3  are  obtained.  In  general,  the  width  of  the  dif- 
fraction pattern  increases  without  changing  the  size  of  fringes,  as  the  width 
of  the  available  wave-front  decreases.  A  similar  result  will  be  described  in 
connection  with  figure  48,  Chapter  II.  Naturally,  if  the  displacement  is 
considerable,  it  is  accompanied  by  some  rotation  of  fringes. 

15.  Displacement  parallel  to  rays.— It  now  becomes  of  importance  to  test 
the  range  of  displacement  as  modified  by  the  angle  of  reflection,  increasing 
from  5  =  o.  It  is  therefore  desirable  to  make  a  few  direct  measurements.  The 
angle  0  at  P,  figure  14,  was  found  to  be  about  49°  45',  so  that  the  total  angle 
at  M  is  8  =  40°  1 5'.  M  and  N  are  both  on  micrometers,  with  the  screws  normal 


REVERSED   AND   NON-REVERSED   SPECTRA. 


29 


to  their  faces.  P'  is  on  a  micrometer  with  its  screw  parallel  to  bbf,  so  that 
this  prism  is  shifted  right  and  left.  The  range  of  displacement  was  found  at 

M,  about  0.04  cm.;  #  =  2X0.04X0.939  =  0.076  cm. 
P',  about  y  —  o.o'j  cm.;  2^  =  0.140  cm. 

where  x  =  ze  cos  (90  —  0)/2  and  zy  are  the  corresponding  path-differences 
between  the  inception  and  evanescence  of  fringes.  With  a  very  fine  slit, 
2y  was  possibly  smaller  (see  fig.  17). 

The  question  at  issue  is  thus,  in  the  first  place,  how  the  value  of  zy  compares 
with  oc  ;  for  in  the  former  case  the  angle  5  is  effectively  zero.  In  other  words, 
when  M  is  displaced  from  M  to  M',  over  a  distance  e,  the  pencil  b,  figure  17, 
changes  to  bi,  and  is  soon  lost  at  the  edge  of  Pf,  whereas,  when  P  is  displaced 
in  the  direction  W,  over  a  distance  y,  the  rays  6  and  b'  do  not  change  their 
point  of  impact  at  the  prismatic  mirror  P'.  If  PP  represents  the  principal 
plane  of  the  objective  of  the  telescope  and  F  its  principal  focus,  there  should 
be  no  accessory  effect  for  the  case  y  as  compared  with  the  case  x. 

Results  bearing  on  this  subject  are  given  in  table  9,  in  which  the  displace- 
ment e,  observed  at  M  and  at  N,  as  well  as  the  displacement  y  at  P',  are 
recorded  when  a  plate  of  glass  of  thickness  E  is  inserted  normally  to  the 
rays  b,  b'.  The  corresponding  air-path  difference  computed  from  E,  n,  B, 
and  X  should  be  z,  nearly.  This  is  about  the  value  (zy)  observed,  remembering 
that  to  set  the  micrometer,  fringes  of  a  particular  pattern  must  be  selected. 
The  rotation  of  fringes  being  but  90°  or  less,  there  are  no  fiducial  horizontal 
lines. 


TABLE  9.  —  Reversed  spectra.     Refracting  prism.     6  =  49°  45 
=  2ecos(90°-0)/2;z=E  Oi- 


5=4.6X10-"  (assumed). 


Detail. 

E 

M 

z 

e 

X 

2y 

cm. 

cm. 

cm. 

cm. 

cm. 

cm. 

Mirror  right,  plate  right.  .  . 

0.736 

1.526 

f  0.3901 
1+-OI95 

\ 

0.204 
f  .199 
[  .206 

0.381 

0.400 
.410 

Mirror  left,  plate  left  

.736 

1-526 

=  .4096 

.208 

•391 

f  .410 
\  .406 

Mirror  right,  plate  right.  .  . 

•736 

1.526 

=   .4096 

.201 
.205 
.202 

j  -381 

f  .406 
I  -402 

Mirror  left,  plate  left  

•736 

1.526 

=  .4096 

.207 

.389 

.402 

.207 
.208 



.400 

Mirror  right,  plate  right.  .  . 

•736 

1-526 

=  .4096 

• 

.205 
.207 

}     .387 

.406 

Mirror  left,  plate  left  

•736 

1.526 

=  .4096 

.206 
.208 

]     -389 

.406 

Mirror  right,  plate  right.  .  . 

•434 

1-533 

|     -2313 

• 

•125 

.124 

}     -234 

/  -248 
\  -250 

|  -1-  oi  T  c\ 

.248 

Mirror  left,  plate  left  

•434 

1-533 

IT«V«*3 

=  .2428 

\ 

.126 

.125 

}     -236 

(.255 
•248 
I  -246 

Mirror  left,  plate  right  

•434 

1-533 

=  .2428 

\ 

'  .127 
,  .126 

}    .37 

30 


THE   INTERFEROMETRY   OF 


Furthermore,  though  p  was  determined  by  the  total  reflectometer  for  each 
of  the  two  plates  used,  B,  the  Cauchy  dispersion  coefficient,  had  to  be  assumed 
from  similar  results  in  my  earlier  work.  Finally,  the  first  plate  (£  =  0.736 
cm.)  was  slightly  wedge-shaped  and  some  adjustment  for  coincidence  of 
spectra  was  needed.  The  second  plate  (£  =  0.434  cm.)  was  optically  nearly 
plane  parallel.  One  may  therefore  conclude  from  these  details  that  zy=z  as 
nearly  as  could  be  expected. 

The  values  of  x,  computed  from  8  and  e,  however,  certainly  fall  below  z, 
being  about  6  per  cent  and  3  per  cent  short  of  it  in  the  two  cases,  respectively; 
or,  again,  x  is  0.019  cm-  and  0.014  (about  5  per  cent)  smaller  than  the  mean 
values  observed  for  zy.  This  extra  5  per  cent  of  path-difference  can  not  be 
an  error  of  observation  or  of  adjustment,  but  must  be  interpreted  as  the  path- 
difference  added  when  the  pencil  shifts  towards  the  edge  of  the  prism  (x) 
instead  of  being  stationary  as  in  y.  In  case  of  inverted  spectra,  moreover 
(next  chapter),  x  is  usually  in  excess  of  0,  and  the  shift  is  the  other  way.  The 
deficiency  in  x,  though  not  equally  marked,  is  present  in  observations  both 
on  the  right  and  left  sides  of  the  prism  P1 '. 

16.  Breadth  of  efficient  wave=fronts.  Apparent  uniformity  of  wave=trains. 
Rotation  of  fringes. — It  follows  from  figure  17  that  if  M  is  displaced  to  M't 
over  a  distance  e,  the  pencil  b  is  displaced  parallel  to  itself  over 

5  =  2e  sin  5/2 

where  5  =  90°—  6.  The  pencil  c  is  then  displaced  parallel  to  itself  over  a 
distance 

*  =  stan  <f>'/2=s 

since  6  =  49°  45',  5/2  =  20°  7',  and  therefore  5  =  20X0.344  =  0.70,  nearly. 
If  the  rotation  of  fringes  is  but  90°,  either  5  (or  5/2)  is  also  the  breadth  'of 
the  strips,  or  patches  of  like  origin,  which,  when  sliding  over  each  other  more 


17 


18 


or  less,  produce  the  fringes.    This  may  be  treated  from  a  graphic  point  of 
view  as  follows,  a  theory  not  being  aimed  at : 

In  figure  1 8,  let  a  and  b  be  two  patches  of  light  of  like  color  and  origin  at 
the  objective  pp,  figure  17,  producing  interferences  at  the  focus  F.  Hence 
the  fringes  will  be  arranged  in  the  direction/,  figure  18,  at  right  angles  to  the 


REVERSED  AND   NON-REVERSED   SPECTRA.  31 

line  joining  a  and  b.  Since  a  and  b  here  correspond  to  c  and  c'  in  figure  1 7,  let 
a  be  continually  displaced  to  the  right,  as  indicated  by  the  arrows,  figure  18. 
In  proportion  as  the  positions  ab,  a'b',  a"b",  are  taken,  the  fringes  must  pass 
by  rotation  from  /,  into  /',  into  /",  etc. — i.e.,  over  about  90°.  In  the  present 
experiment,  c,  figure  17,  can  never  pass  across  c' ',  for  they  are  essentially 
separated  by  the  edge  of  the  right-angled  prism  P'.  Hence  the  rotation  can 
not  exceed  90°,  for  the  vertical  through  a  can  not  cross  the  vertical  through  b. 
This  is  not  the  case  when  a  grating  replaces  P',  as  in  figure  12;  nor  is  it 
the  case  when,  as  in  Chapter  II,  inverted  spectra  are  treated,  and  the  patches 
a  and  b  slide  along  the  edge  of  the  prism.  In  such  cases  figure  18  may  be 
continued  symmetrically  toward  the  right  (mirror  images),  and  the  limit  of 
rotation  is  therefore  180°.  All  these  suggestions  are  borne  out  by  experiment. 

Moreover,  if  the  first  prism  P,  figure  14,  is  tilted  slightly  on  an  axis  parallel 
to  LT,  a  (fig.  1 8)  will  be  lowered  and  b  raised.  If  a  and  b  are  on  the  same 
level,  the  fringes  are  always  vertical  and  pass  through  a  vertical  maximum, 
when  ab  is  a  minimum.  On  the  other  hand,  if  a  and  b  are  not  in  the  same 
level,  as  in  the  figure,  fore-and-aft  motion  brings  the  rays  c  and  c'  (fig.  17) 
to  or  from  the  edge  of  the  prism  P'.  Hence  the  case  ab  passes  into  a"b",  or 
the  reverse;  in  other  words,  the  fringes  pass  through  a  horizontal  maximum 
when  ab  is  a  minimum,  etc.  This  is  also  shown  by  experiment. 

Moreover,  if  a,  figure  18,  is  the  angle  (in  the  observer's  vertical  plane)  of 
ab  to  the  horizontal,  the  horizontal  distance  between  c  and  c'  will  be  ab  cos  a, 
which  is  zero  when  a  =  90°,  and  both  c  and  c'  are  at  the  edge.  Suppose  the 
full  breadth  of  the  strips  are  at  the  edge,  so  that  the  fringes  present  the 
strongest,  coarsest,  but  narrowest  field  of  case  2,  figure  16.  Then  if  either 
c  or  c'  retreats  until  the  fringes  vanish,  the  width  of  the  appreciably  effi- 
cient strip  cc'  will  be  ab  cos  a  =  t=s  =  o.'je,  nearly.  This  is  probably  the 
best  method  of  estimating  the  width  in  question.  Usually,  however,  away 
from  the  edge,  the  succession  i,  2,  3,  figure  16,  is  obtained.  In  such  a  case 
the  breadth  of  efficient  strip  is  t/2  =0.350. 

The  experiment  made  by  moving  screens  with  slits,  forward  or  rearward, 
successively,  by  which  the  appearance  and  evanescence  of  fringes  may  be 
repeated  through  several  cycles,  is  next  to  be  explained.  Here  it  is  merely 
necessary  to  remember  that  the  spectra  c  and  c'  are  reversed,  or  that  the 
colors  of  like  origin  and  wave-length  are  successively  farther  apart.  When 
the  screens  are  alternately  moved,  therefore,  the  same  phenomenon  is  in  turn 
produced  in  slightly  different  colors.  But  as  ab  continually  increases,  whereas 
the  efficient  breadth  of  the  strips  does  not,  the  fringes  soon  pass  beyond 
appreciable  smallness. 

When,  as  in  the  earlier  methods,  but  a  single  grating  is  used  with  two 
successive  diffractions  through  it,  the  patches  a  and  b  are  obviously  in  the 
same  level  when  the  longitudinal  axes  of  spectra  coincide.  Hence  the  fringes 
are  essentially  vertical. 

In  the  experiment  with  screens,  5,  s',  figure  14,  it  is  obvious  that  path- 
difference  remains  constant.  The  distance  from  the  same  wave-front  in  the 


32  THE   INTERFEROMETRY  OF 

pencils  b  and  b',  figure  17,  to  the  principal  plane  pp,  is  always  the  same;  but 
pencils  different  in  lateral  position  are  successively  selected.  On  the  other 
hand,  when  the  prism  P'  is  moved  in  the  direction  y,  parallel  to  bb',  path- 
difference  only  is  introduced,  while  the  pencils  selected  remain  the  same. 
Supposing  the  ordinary  conditions  of  visibility  (magnification,  etc.)  to  remain 
unaltered  throughout,  the  wave-fronts  are,  as  it  were,  explored  in  depth  as 
to  their  uniformity — i.e.,  the  distance  is  apparently  recorded  throughout 
which  a  wave-train  consists  of  identical  wave-elements.  Effectively,  however, 
the  rapidity  with  which  fringes  decrease  in  size  beyond  visibility  is  directly 
in  question.  Finally,  when  the  opaque  mirror  M  (or  N)  is  moved  from  M 
to  M',  both  effects  occur  together.  Path-difference  x  =  2e  cos  5/2  is  introduced 
and  the  pencil  is  displaced  from  6  to  b'. 

The  ^-effect  is  thus  probably  the  same  as  if  P'  were  displaced  to  the  left 
and  5  were  brought  forward.  Hence  it  is  of  great  interest  to  determine  the 
extent  in  which  the  values  of  y  and  x  are  different.  It  appears  as  if  the  dis- 
tance within  which  the  wave-trains  are  uniform  is  definitely  limited  and  that 
it  increases  with  the  breadth  of  effective  wave-front  just  instanced,  while 
both  increase  with  the  amount  of  dispersion  to  which  the  incident  white 
pencil  has  been  subjected.  Diffraction  at  the  slit  of  the  collimator  may  be 
regarded  as  the  first  dispersion.  This  seems  to  me  to  be  a  very  important 
observation,  and  a  systematic  investigation  of  the  lengths  of  uniform  wave- 
trains,  so  understood,  in  their  dependence  on  dispersion,  is  desirable,  even  if 
the  geometry  of  the  system  should  prove  to  be  adequate  to  explain  the 
phenomena. 

17.  Film  grating. — The  method  of  two  gratings  was  now  again  resorted  to, 
except  that  the  first  at  G,  figure  3,  was  a  film  grating.  This  attempt  failed 
in  my  earlier  work,  when  but  a  single  film  grating  was  used  for  the  two  dif- 
fractions, because  of  insufficient  light.*  In  the  present  case,  where  two 
gratings  (G1  being  reflecting)  are  employed,  the  method  succeeded  at  once. 
The  first  grating  constant  was  D  =  io^X  167  cm. ;  observations  were  therefore 
necessarily  made  in  the  second  order  of  G',  so  that  the  spectra  are  not  as 
intense  as  with  prisms.  But  the  fringes  are  perfect  and  may  be  made  as  large 
as  desirable— with  but  two  in  the  breadth  of  the  spectrum,  for  instance.  They 
come  in  and  go  out  of  range  with  inflation  of  form,  and  they  are  free  from  the 
awns  seen  in  the  preceding  paragraph  (with  prism),  probably  because  the 
light  is  less  intense. 

The  phenomena  in  general  are  the  same  in  character;  but  the  range  of 
displacement  of  either  mirror  is  enhanced,  conformably  with  the  increased 
dispersion  at  G.  This  range  was  found  to  be  about  6  mm.  under  the  best 
conditions  (arrows).  If  both  M  and  N  are  successively  displaced  in  the 
same  direction,  the  total  displacement  available  between  the  hair-like  fringes 
at  the  extremes  is  about  1.5  cm.  for  each  mirror. 

*  I  have  since  obtained  the  fringes  with  a  single  film  grating. 


REVERSED   AND   NON-REVERSED   SPECTRA. 


33 


At  these  extremes  the  two  patches  of  light  on  the  grating  G'  may  have 
been  separated  by  several  millimeters.  The  nature  of  the  transformation 
from  arrows  to  the  oblique  striations  would  be  well  reproduced  if  equidistant 
vertical  wedges  were  moved  from  right  to  left,  or  the  reverse,  behind  a 
vertical  slit. 

The  distance  between  G  and  G'  was  about  10.2  cm.,  and  between  the  spots 
of  light  on  mirror  M  and  gratings  G  and  G',  respectively,  22.2  and  12.6  cm. 
This  corresponds  to  01  =  20.5° and  #2  =  36. 2°,  so  that  6  =  15.7°  and  <r=s6.7°. 

It  is  obvious  that  if  the  slit  of  the  collimator  is  displaced  right  or  left,  the 
range  of  displacement,  within  which  the  interferences  lie,  will  have  different 
positions  on  the  micrometer,  because  the  path-differences  are  changed.  A 
flickering  arc  may  also  introduce  annoyances. 

The  present  method  has  an  advantage  for  ordinary  practical  purposes,  as 
it  does  not  require  a  ruled-glass  grating  at  G. 


20 


The  surprising  success  obtained  with  the  film  grating  at  short  distances 
induced  me  to  test  similar  methods  at  long  distances.  Figure  19  is  an  appa- 
ratus of  this  kind,  in  which  L  is  the  white  beam  incident  from  a  collimator, 
G  and  G'  are  the  transmitting  gratings,  M,  N,  m,  n,  pairs  of  opaque  mirrors, 
T  the  telescope.  The  undeviated  ray,  d,  is  screened  off.  The  component 
paths  a+b+c,  a'+b'+c'  were  each  about  4  meters  long.  The  method  of 
adjustment  again  consisted  in  bringing  the  shadow  of  the  thin  wire  across 
the  slit  into  the  same  position  of  the  spectra  seen  in  the  telescope  when  the 
spectra  coincide.  For  this  purpose  the  adjustment  screws  for  horizontal  and 
vertical  axes  on  M,  N,  m,  n  must  be  actuated  together.  To  facilitate  this 
tiresome  work,  with  the  observer  at  T,  long  levers  brought  from  m  and  n, 
with  their  ends  near  his  hands,  as  well  as  a  lever  from  G'  (fore-and-aft  motion) , 
were  very  useful.  Since  the  adjustment  screws  at  M  and  N  are  already 
within  reach,  it  is  thus  easy  to  bring  any  Fraunhofer  line  to  the  middle  of  the 
field  and  to  make  these  fields  overlap,  with  the  guide- wire  central  in  both. 

The  attempt  made  with  sunlight,  to  find  the  fringes  when  both  G  and  G' 
are  film  gratings  (D  =  167  X  io-«  cm.),  did  not  succeed.  The  light,  moreover, 


34  THE   INTERFEROMETRY  OF 

is  not  as  bright  as  desirable,  owing  to  the  strong  dispersion.  When  the  grating 
G'  was  replaced  by  a  ruled-glass  grating  (D  =  3 52X10^  cm.),  the  dispersion 
was  not  much  reduced,  but  the  light  was  better.  The  fringes  were  now  found 
after  some  searching  and  seemed  to  be  of  DiD2  breadth,  a  strip  of  oblique 
lines  of  the  usual  character.  But  they  were  not  brilliant  and  were  hard  to 
recover  when  lost.  The  Fraunhofer  lines  were  still  disagreeably  blurred. 

On  exchanging  the  gratings  (ruled-glass  grating  at  G  and  film  at  G'),  though 
the  dispersion  was  smaller,  the  brilliancy  of  spectra  was  greatly  improved. 
The  fringes  came  out  fairly  sharp.  However,  on  cutting  down  the  incident 
beam  at  the  collimator  and  near  G  to  a  breadth  of  not  more  than  0.5  cm.,  the 
fringes  were  acceptable  and  capable  of  high  magnification.  They  remained 
visible  for  a  displacement  of  5  mm.  at  the  micrometer  at  M.  With  fore-and- 
aft  motion  of  G',  the  fringes  rotated  as  usual  from  fine  vertical  hair-lines, 
through  the  horizontal  (probably  arrow-shaped  forms  of  maximum  size), 
back  again  to  hair-lines.  Here  the  excursion  of  G'  was  about  1.5  cm.  On 
tilting  the  grating  G'  in  its  own  plane  and  readjusting  M,  the  rotation  is 
through  the  vertical  maximum  (the  linear  phenomenon).  With  a  slotted 
screen  (0.5  cm.)  at  the  collimator,  the  slit  may  be  widened  until  the  Fraun- 
hofer lines  just  vanish.  If  the  slit  is  but  0.2  cm.,  the  fringes  become  bulky 
and  the  play  at  M  is  but  2  mm. 

The  film  grating  may  be  used  by  reflection,  on  adapting  the  apparatus  in 
figure  12  for  this  purpose,  by  supplying  a  ruled  grating  or  prism  at  P  and  the 
film  grating  (with  its  ruled  side  toward  P)  at  G.  If  a  ruled  grating  is  put  at  P, 
the  spectra  and  fringes  are  good;  but  naturally  there  is  deficient  illumination. 
Nevertheless  a  strong  telescope  may  be  used  and  a  range  of  displacement  of 
4  mm.  at  M  is  available.  This  may  be  increased  indefinitely  by  using  a 
micrometer  at  M  and  N  alternately.  The  chief  difficulty  was  the  (incidentally) 
unequal  brightness  of  spectra. 

Again,  the  method  of  figure  14,  apart  from  the  drawbacks  to  which  that 
method  is  incident,  succeeds  almost  perfectly,  both  in  the  first-  and  second- 
order  spectra.  The  fringes  are  strong  and  clear.  An  Ives  grating  of  high 
dispersion  (D  =  167  X 10-*  cm.)  was  tested. 

The  method  of  figure  20,  with  auxiliary  mirrors  m  and  n  to  accommodate 
the  dispersion  of  G,  was  also  successfully  tried.  Here  G  was  originally  a 
concave  reflecting  grating.  It  was  replaced  by  a  film  grating  used  as  a  reflect- 
ing grating,  with  entire  success.  The  ruled  side  of  the  film  should  be  free 
(without  cover-glass),  but  the  reversed  side  cemented  on  plate-glass  as  usual 
and  the  latter  placed  towards  the  telescope  at  T.  The  prism  P,  in  other 
words,  admits  an  abundance  of  light,  so  that  even  the  loss  in  reflection  from 
the  film  is  not  serious.  Sunlight  should  be  used  without  a  condensing  lens ; 
or,  if  the  latter  is  added,  the  light  leaving  the  telescope  is  to  be  narrowed 
laterally. 

18.  Non-reversed  spectra.— The  prismatic  method  of  cleaving  the  incident 
beam  of  white  light  is  available  for  the  superposition  of  non-reversed  spectra, 


REVERSED   AND   NON-REVERSED   SPECTRA. 


35 


under  conditions  where  the  paths  of  the  component  rays  may  have  any 
length  whatever.  It  is  thus  an  essential  extension  to  the  method  (fig.  21) 
given  in  the  preceding  report  (PP't  prisms;  M,  N,  mirrors;  Gp,  Ives  prism 
grating;  T,  telescope),  where  the  path-differences  were  essentially  small  and 
the  spectra  reversed. 

In  figure  22,  P  is  the  first  prism  cleaving  the  white  beam  L,  diffracted  by 
the  slit  of  the  collimator.  M  and  N  are  the  opaque  mirrors,  the  former  on  a 
micrometer.  For  greater  ease  in  adjustment,  the  second  prism  Pf  is  here 
right-angled,  though  this  is  otherwise  inconvenient,  since  the  angle  8  =  90°  —  <p 
is  too  large.  The  rays  reflected  from  P'  impinge  normally  on  the  reflecting 
grating  G  (D  =  200X10^)  and  are  observed  by  a  telescope  at  T.  P,  P',  M, 
and  N  are  all  provided  with  the  usual  three  adjustment  screws.  P'  must  be 
capable  of  being  raised  and  lowered  and  moved  fore  and  aft.  The  field  is 


21 


22 


brilliantly  illuminated.  When  the  path-difference  is  sufficiently  small,~the 
fringes  appear  and  cover  the  whole  length  of  superposed  spectra  strongly. 
They  are  displaced  with  rotation  if  M  is  moved  normally  to  itself. 

As  first  obtained,  the  fringes  were  too  closely  packed  for  accurate  measure- 
ment. But  the  following  example  of  the  displacement  e  of  the  mirror  M, 
for  successions  of  40  fringes  replacing  each  other  at  the  sodium  lines,  shows 
the  order  of  value  of  results:  iose  =  1.55,  1.40,  1.60,  1.55  cm.,  so  that  per  fringe 

86  =  39X10-*  cm. 
The  computed  value  would  be  (<p,  the  prism  angle) 


8e  = 


58.93 


36. 4X10-*  cm. 


2  cos  5/2       2X.8i 

assuming  5  =  90°—^.  The  difference  is  due  both  to  the  small  fringes,  which 
are  difficult  to  count,  and  to  the  rough  value  of  8.  The  range  of  measurement 
is  small  (if  M  only  moves),  not  exceeding  1.6  mm.  for  a  moderately  strong 


36  THE   INTERFEROMETRY  OF 

telescope.  But  one-half  of  this  displacement  is  available,  as  the  fringes 
increase  in  size  (usually  with  rotation)  from  fine  vertical  hair-lines  to  a  nearly 
horizontal  maximum,  and  then  abruptly  vanish.  This  is  one-half  of  the 
complete  cycle. 

If  we  regard  the  component  beams,  a  b  c  and  a'b'c',  as  being  of  the  width 
of  the  pencil  diffracted  by  the  slit  of  the  collimator,  it  is  clear  that  the  maxi- 
mum size  of  fringes  will  occur  when  c  and  c'  are  as  near  together  as  possible ; 
furthermore,  that  as  M  moves  toward  P',  c  continually  approaches  c',  until 
b  drops  off  (as  it  were)  from  the  right-angled  edge  of  the  prism  P'.  To  get 
the  best  conditions — i.e.,  the  largest  fringes — c  must  therefore  also  be  moved 
up  to  the  edge  of  P  and  very  sharp-angled  prisms  be  used  at  both  P  and  P'. 
The  largest  fringes  (lines  about  10  times  the  DiDz  distance)  obtained  with 
the  right-angled  prism  were  often  not  very  strong,  though  otherwise  satis- 
factory. Much  of  the  light  of  both  spectra  does  not  therefore  interfere,  being 
different  in  origin. 

Results  very  similar  to  the  present  were  described  long  ago*  and  found 
with  two  identical  half -gratings,  coplanar  and  parallel  as  to  rulings,  etc., 
when  one  grating  was  displaced  normally  to  its  plane  relative  to  the  other. 
The  edges  of  the  two  gratings  must  be  close  together,  but  even  then  the 
fringes  remain  small  and  the  available  paths  also.  Strong,  large  fringes,  but 
with  small  paths,  were  obtained  by  the  later  method  t  of  two  identical  trans- 
mitting gratings,  superposed. 

If  the  prism  P'  is  right-angled  (a  special  case  of  fig.  21),  it  may  be  rotated 
as  in  figure  23,  so  that  the  rays  c  and  c'  pass  off  towards  the  observer.  They 
are  then  regarded  through  a  prism-grating  G  and  a  telescope  at  T.  This 
method  admits  of  much  easier  adjustment.  With  the  component  beams 
a  b,  a'b' ,  coplanar,  horizontal,  and  of  about  equal  length  in  the  absence  of 
the  prism  P',  the  latter  is  now  inserted  with  its  edge  vertical  (rotation)  and 
the  white  slit  images  in  T  (without  G]  superposed,  horizontally  and  vertically. 
G  is  then  added  and  the  micrometer  at  M  or  AT  manipulated  till  the  fringes 
appear.  As  above,  they  are  largest  when  c  and  c'  are  as  nearly  as  possible 
coincident  and  vanish  as  horizontal  fringes  at  the  maximum;  for  the  effective 
parts  of  c  and  c'  are  component  halves  of  the  same  diffracted  beam  from  the 
slit.  It  is  interesting  to  observe,  seeing  that  interference  also  occurs  when 
one  of  the  superposed  spectra  is  inverted  on  a  line  parallel  to  its  length,  that 
such  diffraction  is  demonstrable  in  case  of  homogeneous  light,  even  when  a 
slit  is  absent.  Both  beams  must  be  nearly  at  the  edge  of  P'  in  order  that 
strong,  large  fringes  may  be  seen. 

The  case  of  figure  22  was  subsequently  again  tried  on  the  large  interferom- 
eter, the  distance  P  to  M—N  being  about  2  meters.  G,  in  these  experiments, 
was  a  concave  grating  and  T  a  strong  lens  near  the  principal  focus  of  G.  The 
adjustment  for  long  distances  is  not  easy.  The  equilateral  triangle  of  rays, 
a,  a',  b',  6,  should  be  first  carefully  leveled,  the  edges  of  P  and  P'  being  on 

*Phil.  .Mfg.,  xxn,  pp.  118-129,  19";  Carnegie  Inst.  Wash.  Pub.  149,  chap.  VI. 
Physical  Review,  vii,  p.  587,  1916;  Science,  XLII,  p.  841,  1915. 


REVERSED   AND   NON-REVERSED   SPECTRA. 


37 


the  median  line.  With  G  placed  at  the  proper  distance,  the  two  spectra  seen 
at  T  will  usually  be  quite  distinct  in  the  field.  They  should  show  the  shadow 
of  the  black  line  across  the  slit,  at  the  same  level  in  the  spectra.  The  longi- 
tudinal axis  of  the  spectra  may  then  be  made  collinear  by  slightly  tilting  the 
edge  of  P'  to  the  vertical,  on  a  horizontal  axis,  with  the  adjusting  screws. 
M  and  N  are  then  rotated  on  a  vertical  axis  till  the  D  lines  coincide.  Small 
changes  may  be  completed  at  M  and  N.  The  fringes  when  found  are  usually 
strong,  but  very  fine,  less  than  the  D\D%  distance  in  width.  I  have  been  able 
to  increase  them  to  a  width  of  2DiDz,  but  they  are  then  faint.  The  two 
illuminated  strips  on  the  grating  may  even  be  an  inch  apart,  but  the  fringes 
are  as  usual  large  when  this  distance  is  the  smallest  attainable  (virtual 


24 


23 


coincidence).  The  grating  may  be  moved  fore  and  aft  without  effect.  As 
N  is  moved  on  its  micrometer,  the  interferences  are  first  seen  as  vertical 
hair-like  striations,  which  gradually  enlarge,  rotate,  and  vanish  just  before 
reaching  the  horizontal  and  at  maximum  size.  The  range  of  displacement  did 
not  exceed  0.15  cm.  for  this  rotation  of  90°,  so  that  the  total  displacement 
for  1  80°  of  rotation  would  be  about  3  mm.  Since  N  and  T  are  close  together, 
the  manipulation  is  convenient  here,  but  with  another  lens  at  T'  the  phenom- 
enon could  be  traced  further  on  the  M  side. 

To  secure  a  smaller  angle  of  incidence  and  reflection,  5/2,  at  M,  figure  24, 
the  combination  of  a  silvered  20°  prism  P  and  a  30°  prism  P'  was  tested.  M 
and  N  are  the  opaque  mirrors,  G  the  concave  grating  with  its  focus  at  T  for 
inspection  by  a  strong  lens.  L  is  the  incident  beam  of  white  sunlight  from  the 
collimator,  which  is  split  into  the  component  pencils  abed  and  a'b'c'd'  and 
interfere  at  T.  The  results,  however,  were  about  the  same  as  above,  the 
range  of  displacement  at  M  for  90°  of  rotation  of  fringes  being  about  0.15  cm. 
As  a  and  b  make  angles  <p  and  <?'  with  the  line  of  symmetry  LL't 


was  about  10°. 

At  a  subsequent  opportunity  I  made  further  trials  with  the  paired  prisms 
of  20°  and  30°,  but  failed  to  increase  the  fringes  above  about  D\Dt/2  width. 


38  THE   INTERFEROMETRY  OF 

Two  micrometers,  one  at  M  and  the  other  at  N,  were  installed,  and  moved 
forward  in  alternate  steps,  within  a  range  of  over  2  cm.,  naturally  without 
modifying  the  fringes.  These  are  now  observed  on  both  sides  (N an d  M) ,  each 
with  the  micrometer  which  is  manipulated.  One  may  note  in  passing  that  the 
two  screws  are  being  incidentally  compared.  To  set  the  30°  prism  properly  it 
would  have  to  be  provided  with  a  fine  fore-and-aft,  right-and-left  slide  adjust- 
ment, in  order  that  its  edge  may  be  set  sharply  in  the  line  where  the  two 
component  rays  intersect.  An  attempt  was  made  to  increase  the  dispersion 
by  allowing  a  spectrum  to  fall  on  the  first  prism  (20°),  but  without  success. 

It  is  noteworthy  that  the  30°  prism  at  P'  is  no  marked  improvement  as 
to  range  of  displacement  over  the  90°  prism  at  P',  previously  used.  In  other 
words,  the  effect  of  decreasing  the  angle  of  reflection  5  at  M  is,  unexpectedly, 
of  small  importance,  in  relation  to  the  range  of  displacement  at  M.  This  result 
already  treated  in  §16  will  be  accentuated  in  other  ways  below  (Chapter  II). 

19.  Non=re versed  spectra.  Restricted  coincidence. — In  figure  25,  the  white 
ray  L  from  the  collimator  is  diffracted  by  the  grating  G  and  the  two  spectra 
a  and  a',  thereafter  reflected  by  the  parallel  opaque  mirrors  M  and  N,  to 
be  again  diffracted  by  the  grating  G'.  The  rays  are  observed  by  a  telescope 
at  T.  If  the  gratings  G,  G'  have  the  same  constant,  it  is  obvious  that  the  field 
of  the  telescope  will  show  a  sharp  white  image  of  the  slit,  for  each  mirror. 
If  M  N  G  G'  are  adjusted  for  symmetry  by  aid  of  the  adjustment  screws  on 
each  and  the  rulings  are  parallel,  the  two  white  slit  images  will  coincide 
horizontally  and  vertically.  If  now  a  direct-vision  spectroscopic  prism  or 
a  direct-vision  prism-grating  G"  is  placed  in  front  of  the  telescope,  the  super- 
posed white  slit  images  will  be  drawn  out  into  overlapping  non-reversed 
spectra,  which  will  usually  show  a  broad  strip  of  interference  fringes. 

In  my  first  experiments,  G  and  G'  were  film  gratings  of  high  dispersion 
(D=i67Xio-6).  The  field  was  therefore  too  dark  and  the  fringes  were 
obtained  with  difficulty  because  of  the  cumulative  distortion  of  images  from 
the  gratings.  When  found,  the  fringes  were  very  fine  parallel  lines,  filling 
an  irregular  strip  in  the  orange-yellow  region,  and  it  was  already  obvious 
that  an  enormous  play  of  the  micrometer-screw  at  M  would  be  permissible. 

A  number  of  film  gratings  were  now  tested  and  the  best  samples  selected 
(£  =  175X10-*),  although  the  dispersion  was  still  too  large  and  the  D  lines 
not  clear.  To  secure  more  light,  a  beam  of  sunlight  15  cm.  in  diameter  was 
condensed  to  a  focus  on  the  slit  by  a  large  lens  of  about  2  meters  in  focal 
distance.  The  illumination  was  now  adequate  and  the  fringes  were  found 
at  once,  as  they  should  be;  for  they  are  always  present,  though  not  in  the 
same  colored  region.  These  fringes,  found  with  more  accurate  adjustment, 
were  also  larger  than  before. 

Figure  25  shows,  if  ab,  a'b'  and  cd,  c'd'  are  pairs  of  corresponding  rays  of 
the  same  order  but  different  wave-lengths,  X  and  X'  respectively;  that  for 
the  given  position  of  G  and  G'  only  the  rays  a  a'  issue  coincidently  at  T. 
The  rays  cd,  c'd'  issue  at  eit  e\,  and,  though  brought  to  the  identical  focus 


REVERSED  AND   NON-REVERSED   SPECTRA. 


by  the  telescope,  the  distance  eie\  may  be  too  large  to  admit  of  appreciable 
interference.  Hence  the  colored  strip  within  which  interferences  occur  will 
comprise  those  wave-lengths  which  lie  very  near  X,  whereas  the  colors  lying 
near  X',  etc.,  will  be  free  from  interference. 

If,  however,  the  mirror  M  is  displaced  parallel  to  itself  to  M'  by  the  microm- 
eter-screw, the  rays  c"d"  and  c'd'  will  now  coincide  at  e'\,  whereas  the  rays 
from  ab  and  a'b'  will  no  longer  issue  coincidently  and  may  not  interfere. 
Thus  the  interferences  are  transferred  as  a  group  from  rays  lying  near  X  to 
rays  lying  near  X'.  It  is  obvious,  therefore,  that  with  the  displacement  of  M 
the  strip  carrying  interferences  will  shift  through  the  spectrum  and  that  an 
enormous  play  of  the  micrometer-slide  at  M  will  be  available  without  the  loss 


26 


of  interferences.  In  fact,  a  displacement  e  of  over  3  cm.  of  M  normal  to  itself 
produced  no  appreciable  change  in  the  size  or  form  of  fringes,  but  they  passed 
from  the  green  region  into  the  red.  In  consequence  of  the  film  gratings  used, 
the  strip  in  question  was  naturally  sinuous  and  somewhat  irregular,  but  the 
fringes  themselves  in  the  clear  parts  were  straight,  parallel ,  strong  lines.  They 
did  not  thin  out  to  hair-lines  at  their  ends,  nor  show  curvature,  as  one  would 
be  inclined  to  anticipate.  On  the  contrary,  they  terminated  rather  abruptly 
at  the  edges  of  a  strip  occupying  about  one-fourth  of  the  visible  length  of 
the  spectrum. 

It  follows,  therefore,  from  figure  25,  that  the  displacement  of  Mdoes  not 
change  path-length  or  path-difference,  for  the  rays  are  inclosed  between 
parallel  planes,  as  it  were.  Since  the  double  angle  of  reflection  is  8  =  iSo°—2d, 
where  0  is  the  angle  of  diffraction  of  G  and  Gf,  the  displacements  of  M 
over  a  normal  distance  e  will  shorten  the  path  of  M  in  accordance  with  the 
equation 
(i)  nX  =  2e  cos  5/2  =  2e  sin  6 


40  THE   INTERFEROMETRY  OF 

where  n  is  the  number  of  fringes  passing  at  wave-length  X.  This  equation 
is  not  obvious,  since  for  constant  X,  the  distance  between  G  and  G',  measured 
along  a  given  ray  (prolonged)  for  any  position  of  M  or  N,  is  also  constant. 
The  equation  may  be  corroborated  by  drawing  the  diffracted  wave-front, 
which  cuts  off  a  length  26  sin  6  from  d". 

Since  sin  Q=\/D  if  D  is  the  grating  space,  the  last  equation  becomes 

n  =  2e/D 
or  per  fringe 

de=D/2 

a  remarkable  result,  showing  that  the  displacement  of  the  mirror  M  per 
fringe  is  independent  of  wave-length  and  equal  to  half  the  grating  space.  An 
interferometer  independent  of  X  and  available  throughout  relatively  enormous 
ranges  of  displacement  is  thus  at  hand.  It  will  presently  appear  that  it  is 
also  independent  of  the  angle  of  incidence  at  G. 

In  case  of  the  given  grating  and  sodium  light,  8=  19°  37'.  Hence  if  8e  is 
the  displacement  per  fringe, 

<?=X/2  sin  8=  icr'XSS  cm. 

Actuating  the  micrometer  at  M  directly  by  hand  (this  to  my  surprise  was 
quite  possible  without  disturbing  the  fringes,  except  for  the  flexure  of  sup- 
ports), the  following  rough  data  were  successively  obtained  from  displacements 
corresponding  to  10  fringes: 

io6X6e  =  65     95     90    80    60  cm. 

Without  special  precaution  the  fine  fringes  can  not  be  counted  closer  than 
this,  so  that  the  data  are  corroborative. 

If  the  incidence  is  at  an  angle  i,  as  in  figure  26,  the  rays  entering  at  G 
obviously  leave  at  the  same  angle  i  (symmetrically)  at  G'.  In  other  words, 
rays  enter  and  leave  in  pencils  of  parallel  rays.  The  optic  path  of  the  com- 
ponent N  ray,  /+g,  is 

h/cos  8" 

where  h  is  the  normal  distance  between  G  and  G'  and  6"  the  angle  of  diffrac- 
tion on  the  left.  Similarly  the  optic  path  of  the  M'  rays  c-\-d  which  meet 
/+g  in  r  is 

h/cos  6' 

where  0'  is  the  angle  of  diffraction  on  the  right.  If  now  M'  is  displaced  to  M, 
c+d  is  changed  to  a+d  of  the  same  length ;  but  if  the  wave-front  w  is  drawn, 
it  appears  that  the  optic  path  of  the  M  ray  has  been  shortened  to  20  sin  6', 
where  e  is  the  normal  displacement  of  M'  to  M.  Hence  the  path-difference 
is  now 

h/cos  6'— 2e  sin.  e'— h/cos  6"  =  n\ 

n  being  the  ordinal  number  of  fringes  of  wave-length  X.  Furthermore,  since 
h,  i,  0',  and  6"  do  not  change,  the  displacement  per  fringe  is  as  above 

&?  =  X/2  sin  8' 


RESERVED  AND   NON-REVERSED   SPECTRA.  41 

For  the  angles  in  question  the  usual  equations  are  given: 

sin  0' — sin  i = X/D  sin  6" + sin  i = X/D 

so  that 

5<?=X/2(X/.D-fsin;) 

which  is  thus  not  quite  independent  of  X  unless  i  is  very  small.  It  is  obvious 
that  the  optic  paths  of  a + d  and  c +k  are  identical.  Hence  the  path-differences 
of  the  rays  r,  s  are  the  same.  If  now  the  grating  G'  is  shifted  to  G'\,  over  the 
distance  e',  the  same  path-length  is  cut  off  from  both  r  and  s,  and  hence  the 
fringes  do  not  move.  The  locus  or  strip  of  fringes,  however,  is  displaced  in 
wave-length,  bodily,  as  shown  in  figure  25. 

The  equation  in  «X,  which  may  be  written  (d,  a  differential  symbol) 

wX  =  fcS(i/cos  0)-2*  (X/Z?+sinO 

suggests -the  phenomena  to  be  expected  both  when  X  is  constant  and  i  varies 
and  when  i  is  constant  and  X  varies.  The  former  require  a  wide,  the  latter 
a  narrow  slit. 

Some  time  after,  with  an  improved  micrometer,  not  directly  manipulated 
by  hand,  I  obtained  the  following  data  from  a  succession  of  50  small  fringes 
(arc  lamp) : 

eXio3=  3.7     3.9    4.0    4.1     4.2  cm 
5*Xioe  =  74      78      80     81      84     cm. 

Again,  from  successions  of  30  large  fringes: 

eXio~3  =   2.6     2.6     2.4  cm. 
deX  10-^  =  87      87      79      cm. 

All  of  these  are  below  the  value  computed  for  sodium  light,  from  imperfect 
adjustment.  The  march  of  values  in  the  first  series  is  probably  incidental, 
for  I  was  not  able  to  eliminate  the  effects  of  flexure  in  my  improvised 
apparatus.  Again,  the  precise  symmetry  of  the  apparatus  is  not  guaranteed. 

Simple  as  the  method  appears  in  figure  25,  it  is  in  practice  quite  difficult 
to  control.  Fringes  may  be  lost  and  thereafter  hard  to  find  again,  and  this 
in  spite  of  the  large  range  of  displacement.  The  cause  was  eventually  located 
in  the  circumstances  under  which  the  incident  pencil  L  strikes  the  grating  G. 
If  L  shifts  to  right  or  to  left  the  symmetrical  rhombus  of  figure  25  will  be 
converted  into  a  non-symmetric  rectangle  or  into  a  figure  as  in  figure  26. 
If  G  and  G',  M  and  N  were  rigorously  parallel  this  should  not  produce  any 
effect;  but  as  they  are  not  and  as  the  surfaces  are  not  optically  flat  (film 
gratings)  the  effect  is  very  marked  and  probably  of  the  same  nature  as  a 
rotation  of  G'  on  an  axis  normal  to  its  face.  It  requires  but  slight  displace- 
ment of  L  to  right  or  left  to  make  fringes  in  the  yellow  change  to  hair-lines 
in  the  green  or  the  red;  or  they  may  even  be  lost  altogether.  These  fine 
fringes  may  sometimes  be  enlarged,  at  other  times  made  smaller,  by  adding 
or  thickening  (rotation)  the  compensator.  Naturally  in  all  these  cases  the 
overlapping  spectra  are  perfect.  The  only  method  of  rinding  the  fringes 


42  THE  INTERFEROMETRY  OF 

after  the  parts  are  symmetrically  placed  relatively  to  the  light  is  to  move  L 
successively  and  gradually  toward  one  side  or  the  other  and  to  test  each 
case  with  compensators  i  to  2  mm.  thick,  placed  in  the  b  or  b'  pencils.  It 
would  be  advisable  to  place  the  slit  on  a  right-and-left  micrometer.  When 
found,  however,  the  fringes,  if  reasonably  treated,  are  very  persistent,  strong, 
and  easily  enlarged. 

Finally,  the  fore-and-aft  motion  of  G'  must  produce  a  bodily  shift  of 
fringes,  while  the  strip  as  a  whole  is  displaced  in  mean  wave-length;  for 
figure  25  shows  at  once  that  if  G'  were  displaced  to  G'\,  the  X  rays  bb'  would 
lose  their  coincidence  in  T,  while  that  property  would  now  be  possessed  by 
the  X'  rays,  dd'.  If  G'  is  on  a  fore-and-aft  micrometer,  therefore,  one  might 
suppose  a  second  method  of  interf erometry  to  be  available  in  which  symmetry 
is  retained  throughout  and  (since  the  angle  at  which  the  rays  bb'  meet  is 
5  =  20)  subject  to  the  equation 

wX  =  2e'  cos  6/2  =  2er  cos  B 

where  e'  is  the  displacement  corresponding  to  n  fringes  passing  in  wave- 
length X. 

This  equation,  however,  is  inapplicable,  as  already  explained,  because  the 
pencils  bb'  are  not  reflected,  but  diffracted  into  T.  In  the  symmetrical  appa- 
ratus, therefore,  no  perceptible  motion  of  fringes  is  produced  by  the  fore-and- 
aft  motion  of  G',  because  in  all  cases  the  rays  bb'  meet  the  grating  with  a 
constant  phase-difference.  If  the  phases  are  identical  they  remain  so  while 
G'  is  displaced.  The  strip  of  fringes  as  a  whole,  however,  is  slowly  though 
imperceptibly  displaced  through  the  spectrum,  without  accentuated  motion 
of  the  fringes  within  the  strip.  This  inference  was  tested  by  placing  G'  on 
a  fore-and-aft  micrometer.  Large  displacements  of  the  screw  (fractions  of 
a  centimeter)  shifted  the  strip  from  color  to  color  as  specified.  A  micrometric 
displacement  of  G',  however,  was  unaccompanied  by  any  appreciable  dis- 
placement of  fringes.  On  the  other  hand,  any  flexure  of  the  supports  of  G' 
at  once  produced  a  marked  displacement  of  fringes,  while  from  mere  microm- 
eter displacement  no  measurement  could  be  obtained. 

Equation  (i)  is  of  interest  in  interf  erometry,  in  view  of  the  very  long 
ranges  of  displacement  available.  For  such  purposes  gratings  of  lower  dis- 
persion (preferably  ruled  gratings  or  else  prisms)  should  be  used,  to  obtain 
greater  luminous  intensity  in  the  spectrum.  Of  course,  the  gratings  G  and  G' 
may  have  different  constants,  but  in  such  a  case  GNG'M  will  no  longer  be 
a  rhombus.  Since  for  constant  X  the  ray-lengths  in  figure  25  are  constant 
for  all  positions  of  G  parallel  to  G',  M  parallel  to  N,  large  path-difference  may 
conveniently  be  introduced  by  compensators.  If  a  thin  sheet  of  mica  is  moved 
in  either  the  b  or  b'  pencils,  there  is  a  lively  skirmish  of  fringes,  but  they  do 
not  change  size  appreciably.  A  glass  plate  5  mm.  or  more  thick  placed  in 
both  rays  b  and  b'  and  rotated  produces  the  same  results,  but  the  fringes  move 
more  slowly.  A  plate  2.8  mm.  thick,  with  strong  fringes  horizontal  in  the 
yellow,  if  placed  in  the  b'  pencil  produces  hair-lines  inclined  toward  the  left 
in  the  red;  if  placed  in  the  6  pencil,  hair-lines  inclined  to  the  left  in  the  green, 


REVERSED   AND   NON-REVERSED   SPECTRA.  43 

etc.  In  contrast  with  this,  the  shift  from  red  to  green,  if  produced  without 
compensator  by  the  displacement  of  M,  shows  scarcely  any  change  of  fringes, 
either  as  to  size  or  inclination. 

To  change  the  size  of  fringes  it  is  necessary  to  rotate  the  grating  G'  (rela- 
tively to  G)  on  a  horizontal  axis  normal  to  itself.  They  then  both  rotate 
and  grow  larger,  attaining  the  maximum  of  size  when  the  fringes  are  vertical. 
Fringes  quite  large  and  black,  which  are  naturally  much  more  sensitive  to 
compensators,  may  be  obtained  in  this  way ;  but  the  fringes  are  still  easily  con- 
trolled by  hand .  Limitations  of  the  incident  light  in  breadth,  or  simultaneous 
rotation  of  M  and  N,  produced  no  marked  effects. 

Fringes  may  also  be  enlarged  on  moving  the  collimator  with  slit  micro- 
metrically  right  or  left,  as  already  stated,  though  this  must  be  done  with 
caution,  as  the  effects  are  often  surprisingly  abrupt;  for  when  the  system 
is  not  quite  symmetric  displacements  on  G  will  be  equivalent  to  accentuated 
displacement  on  G',  owing  to  the  reflections.  The  reflected  rays  soon  cease 
to  intersect  and  the  displacement  on  M  and  N  is  invariably  large.  Further- 
more, by  the  insertion  of  compensators  (glass  plates  i  to  2  mm.  thick)  in 
the  b  or  b'  pencils,  either  directly  or  differentially,  larger  or  smaller  fringes 
may  be  obtained. 

It  is  now  of  interest  to  return  to  the  equation  referring  to  the  displacement 
of  Gf,  normal  to  itself,  and  to  consider  the  resolving  power  of  the  system ;  for 
the  latter  bears  a  close  analogy  to  the  experiments  made  in  a  preceding 
paper  (Carnegie  Inst.  Wash.  Pub.  249,  Chap.  V,  1916)  on  the  remarkable 
behavior  of  crossed  rays.  If  G'  is  displaced  to  G'\  over  a  distance  e'=dh  (see 
fig.  25,  where  h  is  the  distance  apart  of  G  and  G'),  the  rays  X7  meeting  in  T 
will  now  be  in  the  same  condition  as  were  originally  the  rays  X.  In  other 
words,  e\  and  e'\  have  become  coincident  at  G'.  If  we  assume  that  the  same 
type  of  fringe  results  in  these  cases,  and  if  X'— X=<£X,  6  —  6'=d8  (for  the 
passage  of  bb'  into  ddr  is  in  the  direction  from  red  to  violet), 

(2)  dd=dh  sin  6  cos  6/h,  nearly 

Since  \=D  sin  6  and  d\=  —  D  cos  B  dd,  this  may  be  changed  to 

(3)  d\/\=dh(i-\*/D*)/h 

when  D  is  the  grating  constant.    This  is  the  expression  used  heretofore. 

In  general  it  is  to  be  noted  that  the  present  method,  apart  from  any  prac- 
tical outcome,  is  of  great  interest  as  to  the  data  it  will  furnish  of  the  width 
of  the  strip  of  spectrum  carrying  interference  fringes  under  any  given  con- 
ditions. For  here  the  spectra  are  not  reversed  or  inverted  and  the  latitude 
of  interference  of  diffraction  throughout  X  is  much  broader  than  in  case  of 
reversed  spectra.  But  for  this  purpose  films  will  not  suffice  and  rigid  refracting 
systems  must  be  devised. 

20.  The  same,  continued.    Homogeneous  light.    Dissimilar  gratings.— To 

show  the  close  relation  of  the  present  experiments  with  one  reflection  to  the 


44  THE   INTERFEROMETRY  OF 

earlier  work  with  crossed  rays  and  two  reflections  (I.e.),  experiments  may  be 
made  with  homogeneous  light.  Accordingly,  the  sodium  arc  with  a  wide 
slit  was  installed  and  the  fringes  found  without  difficulty.  Strands  of  fringes 
with  nodules  were  obtained  as  before .  These  rotated  in  marked  degree  ( 1 80°) 
from  vertical  hair-lines,  through  coarse  vertical  strands  with  horizontal 
nodules,  back  to  vertical  hair-lines  again,  as  either  M  or  G'  was  suitably 
displaced  normally  to  its  plane.  To  shift  the  fringes  of  any  form  into  the 
middle  of  the  wide-slit  image,  a  glass  compensator  in  either  b  or  b'  may  be 
resorted  to,  or  both  M  and  G'  may  be  displaced  together.  Again,  whereas 
the  micrometric  displacement  of  M  produces  a  marked  displacement  of  fringes 
within  the  strip  in  accordance  with  equation  (i),  the  micrometric  displace- 
ment of  G'  leaves  the  fringes  stationary  within  the  strip.  While  the  strands 
and  nodules  were  strong,  the  reticulation  of  fringes  could  not  be  clearly  made 
out,  in  view  of  the  use  of  film  gratings  in  place  of  the  ruled  gratings  used  in 
the  earlier  report. 

In  equation  (4),  D=  169X10-*  cm.,  if  dX/X  =  6Xio-8/6Xio-5=icr3,  h  = 
60  cm.,  #=169X10-*  cm.;  whence  d/t=io~3X6o/(i—  (6o/i6g)2)  =  o.o7  cm., 
nearly.  Thus,  if  with  ruled  gratings  the  fringes  due  to  the  D\  and  Dt  lines 
could  be  separately  recognized,  it  should  be  possible  to  distinguish  between 
them  here  also,  as  the  same  phases  require  a  differential  displacement  of  G' 
of  nearly  a  millimeter.  The  same  result  would  be  recognized  at  M  by  a 
displacement  of  dh  tan  8,  where  0  =  21°  nearly,  being  the  mean  angle  of  dif- 
fraction. The  M  displacement  is  thus  0.07X0.36  =  0.025  cm. 

In  case  of  homogeneous  light  the  prism  grating  G"  is  not  needed  and  much 
more  light  is  available  if  the  telescope  is  used  directly.  The  strands  of  inter- 
ferences, being  on  a  yellow  ground,  are  not  very  strong.  Nevertheless  a  few 
measurements  of  ranges  of  displacement  were  made  by  moving  both  M 
(displacement  e)  and  G'  (displacement  h),  alternately.  The  following  values 
of  e,  h,  and  h  tan  0'  were  found,  the  film  gratings  having  nearly  the  same 
constants : 

£  =  0.5  cm.     &=i.3ocm.     h  tan  6'  =  0.49  cm.      0=19°  37'     0'  =  2o°4o' 
e  and  h  tan  0'  coincide  as  closely  as  may  be  expected,  seeing  that  the  fringes 
in  neither  case  can  be  quite  brought  to  vanish. 

Experiments  were  next  made  with  a  grating  of  less  dispersive  power  (D  = 
352  X 10-*  cm.),  ruled  on  glass  and  a  stretched  film  grating  of  the  same  strength. 
It  was  found,  however,  that  the  long  rhombus  GMG'N  was  very  difficult  to 
control,  owing  to  the  reflection  at  almost  grazing  incidence.  The  spectra 
also  were  not  quite  clear.  The  method  was  therefore  eventually  abandoned, 
as  no  fringes  could  be  found. 

The  trial  was  then  made  with  a  weak  grating  at  G  (#  =  352X10-*  cm.) 
and  a  strong  grating  at  G'  (D  =  167X10-*  cm.).  In  adjustment  the  latter 
naturally  overpowers  the  former  and  two  reversed  spectra  are  seen  in  the 
telescope  (without  prism  grating)  immediately  behind  G'.  Both  spectra  were 
quite  strong  and  sharp.  With  white  light  no  fringes  could  be  found  even  after 
long  trial  and  a  variety  of  adjustments. 


REVERSED   AND   NON-REVERSED   SPECTRA.  45 

The  sodium  arc  was  now  used  with  a  very  wide  slit  and  the  fringes  were 
found  without  difficulty.  They  consisted  as  usual  of  vertical  hair-lines 
rotating  through  a  usually  horizontal  maximum  back  to  vertical  hair-lines 
again,  as  either  M,  N,  G'  were  displaced  normal  to  their  faces  on  their  respec- 
tive micrometers.  These  fringes  are  simply  due  to  either  one  or  the  other 
sodium  line  separately  and  therefore  seen  on  a  yellow  ground  free  from  inter- 
ference. Even  after  this,  with  the  apparatus  in  adjustment  for  homogeneous 
light,  white  light  was  tried  again  in  alternation,  but  no  fringes  appeared. 

With  sodium  light  a  few  measurements  of  the  ranges  of  displacement  were 
made.  If  0  and  6'  are  the  angles  of  diffraction,  5=  180°— (0+0')  and  x  =  2e 
cos  5/2,  when  x  is  the  path-difference  cut  off  at  one  end  by  the  displacement  e 
of  the  mirror  M.  The  displacement  of  G'  being  y,  it  appears  that,  apart 
from  sliding,  e  and  y  tan  0'  should  be  nearly  equal.  The  results  were 

*,  £  =  0.42     0.42  0.45  cm.  0=     9°  39' 

x  =  o.8i     0.81  0.87  cm.  0'=  20°  40' 

y=i.?4  i. 80  cm. 

y  tan  6  =  0.66  0.68  cm.  5  =149°  41' 

It  was  not  practicable  to  make  the  hair-lines  quite  disappear  without  a 
large  excess  of  displacement  in  both  cases.  Even  so  the  difference  of  e  and 
y  tan  0'  is  too  large  to  be  explained  by  such  an  error.  But  the  work  with 
the  present  apparatus  (screws  not  long  enough)  is  not  sufficiently  accurate 
to  make  further  discussion  fruitful.  The  error  will  probably  be  associated 
with  the  oblique  incidence  of  rays  in  case  of  a  wide  slit. 

Very  remarkable  results  were  finally  obtained  with  compensators  of  glass 
plate.  Placed  in  one  or  both  beams  and  rotated  around  a  vertical  axis,  they 
rotate  the  fringes.  This  would  be  referable  to  the  sliding  of  the  ends  of  the 
two  pencils  on  the  grating  G'.  If,  however,  they  are  placed  nearly  normally 
in  one  beam,  they  produce  no  effect  either  of  rotation  or  on  the  size  of  the 
fringes;  but  the  pattern  is  displaced  bodily  across  the  wide  yellow  slit  image. 
Glass  plates  0.2  and  0.5  cm.  were  used.  It  is  not  until  the  thickness  of  plate 
reaches  2  cm.  that  appreciable  thinning  of  the  interference  fringes  occurs 
when  the  plate  is  placed  in  one  beam.  With  optic  plate  this  would  be  an 
excellent  method  for  testing  the  lengths  of  uniform  wave-trains.  Finally, 
with  a  fine  slit  and  coincident  sodium  lines,  the  fringes  could  be  seen  in  the 
presence  of  a  continuous  spectrum  as  marked  dots  on  the  enhanced  sodium 
lines.  But  nothing  could  be  detected  with  non-coincident  lines.* 

21.  The  same,  continued.  Duplicate  fringes. — The  occurrence  of  strands 
and  apparently  duplicated  fringes  has  already  been  suggested  in  the  preceding 
paragraph.  In  further  experiments  definite  results  were  eventually  obtained 
with  sunlight.  These  occur  in  very  great  variety,  but  two  typical  phases 
may  be  accentuated,  given  in  figure  27.  In  the  case  a  the  two  sets  are  more 

*  Such  fringes  were  since  found,  incidentally,  in  the  white  flash  of  a  sodium  arc.  They 
were  very  clear,  but  could  not  be  controlled. 


46 


THE   INTERFEROMETRY   OF 


nearly  parallel,  but  one  is  always  very  large  in  comparison  with  the  other. 
In  case  b  the  difference  in  size  is  even  more  marked  and  the  fringes  are  nearly 
orthogonal.  In  intermediate  cases  fine  large  strands  occur.  These  pass  into 
each  other  continuously;  the  manner  does  not  admit  of  description.  They 
are  seen  best  in  the  principal  focal  plane  and  both  sets  are  about  equally  strong. 

To  obtain  these  fringes  the  adjustment  was  first  carefully  made  with  the 
sodium  arc.  Thereupon  the  arc  was  replaced  by  concentrated  sunlight  and 
fine  fringes  were  recognized  in  the  superposed  spectra  (longitudinal  and 
transverse  axes  coinciding).  These  fine  fringes  were  then  enlarged  both  by 
rotating  the  grating  G'  (fig.  25)  on  its  normal  axis  and  readjusting  M  in  each 
case,  and  by  adding  trial  compensators  in  the  M  or  N  pencils.  A  glass  plate 
3  mm.  thick  gave  the  best  results  and  they  were  very  striking.  The  following 
are  the  chief  characteristics: 

When  the  mirror  M  is  slowly  rotated  on  a  horizontal  axis,  moving  one 
spectrum  vertically,  slightly  over  the  other,  the  fringes  pass  through  all  their 
phases. 

When  M  is  slowly  rotated  on  a  vertical  axis,  which  slides  one  spectrum 
horizontally  over  the  other,  the  fringes  are  displaced,  more  or  less,  bodily 
in  the  spectrum.  Thus  in  the  case  b,  figure  27,  the  D  doublets  are  many 

29 


28 

times  their  own  spacing  (DjZ)2)  apart.  If  the  two  D  doublets  approach  each 
other,  the  fringes  approach  the  D  line  from  larger  wave-lengths  and  vice  versa. 
The  fringes  were  lost  when  the  doublets  crossed  over  each  other. 

Rotation  of  the  compensator  in  the  first  place  moves  the  fringes  as  in  inter- 
ferometry,  as  does  also  the  normal  micrometric  displacement  of  M.  If  this 
motion  requires  readjustment  of  M,  the  range  of  displacement  is  curtailed 
and  the  corresponding  change  of  phase  appears.  In  the  second  place,  the 
compensator,  on  rotation,  traces  the  contours  of  the  curves  by  successively 
accentuating  the  vaguer  parts,  as  will  presently  be  explained. 

The  fore-and-aft  motion  of  G'  also  moves  the  fringes  bodily  through  the 
spectrum  without  marked  change  of  phase.  All  fringes,  whether  produced 
with  or  without  compensators,  are  ultimately  curved  lines. 

The  most  remarkable  results  occurred  on  widening  the  slit.  Supposing 
that  large  strands  were  visible  in  case  of  the  fine  slit,  and  that  this  was  grad- 
ually widened  until  the  slit  width  was  0.5  mm.  or  more,  the  strands  were  found 


REVERSED  AND   NON-REVERSED   SPECTRA.  47 

to  have  coalesced  in  a  way  which  defies  description.  In  their  place  appeared 
a  wide  strip  of  equidistant  parallel  crescents,  as  shown  in  figure  28.  The 
Fraunhofer  lines  had  long  vanished  and  the  appearance  of  the  spectrum  was 
whitish  and  intense.  The  fringes  in  question  are  thus  in  a  measure  achro- 
matic. The  strips  appear  quite  regular  through  the  breadth  of  the  spectrum 
and  its  width  may  be  one-third  of  the  length  of  the  spectrum.  The  fringes 
move  with  the  normal  displacement  of  M  (interferometry)  and  the  range  is 
large  (0.5  cm.  without  adjustment),  provided  M  does  not  require  readjust- 
ment by  rotation.  Simultaneously  the  strip  is  displaced  longitudinally  in 
the  spectrum  in  the  usual  way. 

On  closing  the  slit  the  ellipses  break  up  into  sharp  strands  again  without 
offering  a  systematic  clue  as  to  the  manner  in  which  this  is  done.  The  strands 
usually  trend  more  or  less  vertically  with  two  sharp,  strong  groups,  flanked 
by  one  or  more  weak  groups  on  each  side. 

On  removing  the  condenser  these  crescents  became  more  slender  but  much 
sharper,  so  that  in  spite  of  the  diminished  light  they  could  be  well  seen.  They 
were  then  found  to  be  like  the  approximately  confocal  ellipses  of  displacement 
interferometry,  though  not  subject  to  the  same  laws.  They  embraced  over 
one-third  of  the  visible  overlapping  (green-yellow  through  red)  spectra,  ter- 
minating in  very  fine  hair-lines  on  one  side  but  coarse  lines  on  the  other. 
On  opening  the  slit  to  about  o.i  mm.  the  evolution  was  curious.  With  a 
very  fine  slit  a  relatively  narrow  strip  of  strong  slanting  lines  was  seen  in 
the  yellow.  As  the  slit  widened  these  developed  curvature,  adding  the  more 
slender  complements  of  the  ellipses  on  the  red  side,  until  this  part  of  the  spec- 
trum was  filled  with  confocal  half-ellipses  having  a  transverse  major  axis. 
The  range  of  displacement  of  M  is  practically  indefinite,  depending  simply 
on  the  degree  to  which  the  spectra  overlap ;  3  to  4  cm.  were  tried.  A  hori- 
zontally wide  mirror  at  M  is  needed;  for  the  ellipses  travel  through  the  spec- 
trum and  the  pencil  along  the  mirror,  from  end  to  end.  Both  sides  of  the 
ellipses  may  be  traversed  by  rotating  the  plate  compensator,  which  succes- 
sively accentuates  (in  a  transverse  strip)  a  definite  part  of  their  contours. 
In  this  way  the  thick  apices  or  either  of  the  hair-like  lateral  ends  may  be 
clearly  brought  out.  Thus  the  two  lines  a,  in  figure  28,  limit  the  strong  part 
of  the  ellipses.  When  a  moves  to  right  or  left,  the  hair-lines  appear  more 
and  more  strongly  until  they  terminate,  showing  that  the  inclusive  strip  is 
also  limited. 

To  further  study  this  result,  the  grating*  G'  was  successively  rotated  in 
small  amounts  on  a  normal  axis  with  adjustment  at  M.  It  was  thus 
possible  to  find  both  the  ends  of  the  ellipses,  as  well  as  the  central  parts. 
As  a  result  the  form  figure  29  was  definitely  brought  out.  The  con- 
focal  ellipses  are  extremely  eccentric,  with  very  turgid  apices,  so  that  the 
central  part,  if  in  the  spectrum,  consists  of  transverse  straight  lines.  Motion 
of  M  shifts  the  fringes  to  and  from  the  center  where  they  originate  or  evanesce. 

*  When  nearly  centered,  rotation  of  M  about  a  horizontal  axis  is  also  sufficient  to  com- 
plete the  centering  of  the  ellipses. 


48  THE   INTERFEROMETRY  OF 

The  ellipses  move  as  a  whole  with  M,  without  changing  form  appreciably, 
throughout  the  spectrum;  but  they  move  very  slowly,  quite  differently  in  this 
respect  from  the  round  ellipses  in  displacement  interferometry,  which  are 
extremely  sensitive  to  displacement  of  M.  In  the  present  work  it  may  take 
5  or  10  cm.  at  M  to  pass  the  ellipses  quite  through  the  spectrum.  They  are 
strong  and  fine  in  spite  of  the  film  gratings  used. 

In  the  endeavor  to  explain  these  phenomena  one  may  notice  that  the  main 
features  have  already  been  accounted  for.  As  to  details,  since  the  gratings 
are  films  which  may  act  from  both  sides,  explanations  are  hazardous.  I  do 
not  believe,  however,  that  the  films  (cemented  with  balsam  on  glass  plate) 
had  any  other  discrepant  effect  here  than  to  make  straight  lines  sinuous.  The 
character  of  the  phenomena,  as  a  whole,  is  trustworthy. 

In  the  case  of  the  duplicated  fringes  (fig.  27,  a,  b,  and  the  strands)  seen 
with  a  fine  slit,  the  danger  is  perhaps  greatest.  But  it  appears  to  me  that  the 
coarse  lines  in  figure  27  are  vestiges  of  the  ellipses  of  figure  29,  due  to  a  wide 
slit.  These  are  superimposed  on  special  fringes  resulting  from  the  diffraction 
of  the  narrow  slit.  It  is  difficult  to  conjecture  any  other  cause  of  duplication. 

The  shift  of  ellipses  through  the  spectrum  follows  as  before  from  figure  25. 
Their  occurrence  in  case  of  a  wide  slit  might  be  associated  with  the  equation 


or  with  their  independence  of  wave-length.  They  would  therefore  result 
merely  from  the  obliquity  introduced  by  dispersion.  But  the  presence  of  the 
glass-plate  compensator  militates  against  this.  In  the  peculiar  behavior  of 
the  compensators  when  added  or  rotated  around  a  vertical  axis,  the  dispersion 
of  the  glass  itself  comes  prominently  into  play,  for  the  effect  of  introducing 
a  corresponding  air-path  is  negligible  in  its  effect  on  form.  Thus  the  removal 
of  a  3  mm.  compensating  plate  may  leave  the  fringes  almost  too  small  to  be 
seen,  whereas  the  displacement  of  M  over  3  cm.  produces  but  little  change 
of  form. 

Finally,  the  ellipses  are  developed  in  arc  or  contour  from  left  to  right,  for 
instance,  when  the  slit  is  widened  ;  and  they  vanish  from  right  to  left  as  the 
slit  is  more  and  more  nearly  closed.  The  last  lines  for  a  closing  slit  make  a 
narrow  grid  of  fringes  quite  straight  and  strong. 

22.  The  same.  Prismatic  adjustment.  —  The  60°  prism  has  certain  ad- 
vantages in  experiments  like  the  present,  particularly  when  non-reversed 
spectra  are  to  be  obtained.  Figure  30  is  a  device  of  this  kind,  in  which  P  is 
the  separating  prism  and  P'  the  collecting  prism,  the  beam  of  white  light  L 
from  a  collimator  entering  the  flat  face  normally  on  the  front  side  and  issuing 
normally  on  the  rear  side  at  c  and  c'.  M  and  N  are  opaque  mirrors  parallel 
to  each  other,  G  a  direct-vision  prism-grating.  The  telescope  is  at  T.  The 
reflection  may  be  either  internal,  as  in  the  strong  lines  of  figure  30,  or  it  may 
be  external  on  silvered  faces  of  the  prisms  p  and  £',  the  appurtenances  being 
shown  in  dotted  lines.  In  this  case  the  separated  rays  a,  a',  b,  b'  are  collected 


REVERSED  AND   NON-REVERSED   SPECTRA. 


49 


at  c",  c"',  to  be  joined  in  the  telescope  at  T.  The  internal  reflection  being 
total,  with  the  rays  entering  and  leaving  at  right  angles  to  the  faces  and  re- 
quiring no  silvering  of  the  latter,  I  made  use  of  it  for  the  following  experi- 
ments :  M  and  N  are  on  micrometers  with  the  screw  in  the  direction  normal 
to  their  faces.  P,  M,  N,  P'  must  all  be  adjustable.  After  preliminary  meas- 
urement for  equal  distances,  the  fringes  were  found  without  trouble.  They 
were  strong  but  fine,  beginning  with  vertical  hair-lines  and  gradually  rotating 
as  they  grew  coarser,  till  they  rather  abruptly  vanished.  The  displacement 
of  the  M  mirror  did  not  exceed  0.6  mm.,  nor  the  rotation  30°.  The  spectra 
being  non-reversed,  the  fringes  covered  the  whole  field.  Nevertheless  these 
lines  must  probably  be  regarded  as  arcs  of  circles  or  ellipses  with  enor- 
mously distant  centers.  In  fact,  the  appearance  of  the  whole  of  the  elliptic 
symmetry,  in  the  preceding  experiments  (§  21)  with  gratings,  is  also  to 
be  associated  with  a  slight  difference  of  length  of  two  overlapping  spectra. 
This  is  necessarily  the  case,  since  the  two  gratings  G  and  G',  figure  25,  have 
never  quite  the  same  constant.  The  third  grating  must  therefore  produce 
two  spectra,  the  one  slightly  incremented  and  the  other  decremented  in  length, 
respectively,  as  compared  with  the  case  for  white  light. 


•in 


3\ 


One  would  naturally  suppose  that  the  abrupt  evanescence  of  fringes  was 
due  to  the  escape  of  the  b  beam  at  the  edge  of  the  prism  P';  but  this  is  not 
possible,  as  the  mirror  M  was  traveling  toward  the  rear.  Furthermore,  the 
fore-and-aft  motion  of  the  prism  P'  over  several  millimeters  had  scarcely  any 
effect  on  the  fringes.  This  is  unexpected;  for  the  rays,  c,  c',  are  compelled  to 
approach  or  recede  from  each  other  by  this  motion.  Finally,  the  sodium 
doublets  may  be  moved  at  some  distance  (many  times  their  breadth)  apart 
without  destroying  the  fringes.  They  are  often  most  distinct  when  the  D 
lines  are  not  superposed.  The  same  is  also  true  for  the  longitudinal  axes, 
though  to  a  less  degree. 

These  features  are  therefore  peculiar.  The  rays  c  c'  were  about  5  mm. 
apart.  Unfortunately  the  faces  of  the  prisms  were  optically  inadequate,  so 
that  the  sodium  lines  were  not  sharp.  For  this  reason  no  results  were  obtained 
with  homogeneous  light  and  a  wide  slit. 


50  THE   INTERFEROMETRY  OF 

To  enlarge  the  fringes,  the  prism  P'  may  be  rotated  around  a  horizontal 
axis  parallel  to  LT.  The  fringes  then  also  rotate,  but  the  increase  of  size  so 
obtained  is  usually  not  striking.  Moreover,  no  observable  effect,  either  on 
the  size  of  fringes  or  on  the  range  of  displacement,  is  produced  by  inserting 
compensators  in  one  beam  or  both.  If  M  and  N  are  moved  together  toward 
the  right  or  left,  the  result  is  not  appreciable.  A  great  variety  of  different 
adjustments  showed  a  range  of  displacement,  at  M,  about  the  same  (0.06  cm.) , 
whether  the  patch  of  light  on  the  prism  was  wide  or  narrow.  The  range  of 
fore-and-aft  motion  of  P'  within  which  fringes  are  visible  was  0.52  cm.  They 
vanish  quite  abruptly  when  the  light  is  near  the  edge  of  the  prism,  although 
both  spectra  are  still  strongly  visible.  When  the  light  is  nearer  the  base  of 
the  prism  they  vanish  more  gradually.  Definite  strips  of  white  light  on  both 
sides  of  the  prism,  therefore,  cooperate  to  produce  the  fringes.  The  remainder 
of  the  illumination  is  ineffective.  The  distance  apart  of  c  and  c',  as  modified 
by  fore-and-aft  motion,  curiously  enough,  is  here  without  marked  influence. 
It  is  true,  however,  that  the  largest  fringes  were  obtained  when  the  two  pencils 
of  light  from  M  and  N  coincided  at  the  objective  of  the  telescope,  although 
the  D  lines  were  in  this  case  far  apart.  The  attempt  to  find  a  systematic 
method  for  enlarging  the  fringes  failed,  possibly  because  the  prism  angles 
were  not  quite  identical.  The  striking  contrast  in  the  results  obtained  here 
in  comparison  with  those  of  the  preceding  paragraph,  although  both  methods 
are  essentially  the  same,  is  noteworthy. 

It  is  for  this  reason  that  I  thought  it  desirable  to  test  the  method  in  figure  3 1 , 
which  accomplishes  with  a  prism  what  was  done  in  my  original  experiments 
with  reversed  spectra,  by  the  aid  of  a  grating.  In  the  figure  the  incident 
beam  of  white  light  L  from  a  collimator  strikes  the  60°  prism  at  its  edge,  and 
is  then  refracted  into  the  paired  pencils  a,  a'.  These  are  reflected  normally 
by  the  opaque  mirrors  M  and  N,  again  refracted  by  P  as  each  pencil  nearly 
retraces  its  path.  The  return  beams,  however,  are  given  a  slightly  upward 
trend,  so  as  to  impinge  on  the  opaque  mirror  m  (curved  or  plane).  The  rays 
reflected  from  m,  in  such  a  way  as  to  avoid  the  prism  P,  may  be  reunited  in 
the  focus  F  observed  by  the  lens  T,  or  (if  parallel)  collected  by  a  telescope  at  T. 
In  view  of  the  prism,  the  spectra  are  small  and  reversed,  but  may  be  brought 
to  overlap  at  the  red  ends,  which  are  towards  each  other. 

The  small  dispersion  makes  it  necessary  to  use  a  strong  telescope  if  the 
Fraunhofer  lines  are  to  be  visible  and  the  D  lines  separated.  Usually  the 
two  doublets  will  be  at  a  small  angle  to  each  other,  but  this  does  not  mar 
the  interferences.  When  the  adjustment  has  been  made  symmetrically, 
a  strong  linear  phenomenon  maybe  found  not  differing  in  appearance  from  the 
results  obtained  when  a  grating  was  used  at  P,  figure  31.  When  the  mirror 
M  is  displaced,  however,  the  fringes  appear  in  the  form  of  multiple  vertical 
hair-lines,  which  grow  coarser  until  but  a  single  dark  line  flanked  by  a  bright 
line  is  visible.  With  further  displacement  the  phenomenon  again  vanishes  in 
passing  through  multiple  hair-lines.  This  appearance  of  hair-lines  is  one 
distinguishing  feature;  but  a  much  more  important  result  is  the  small  range 


REVERSED  AND   NON-REVERSED   SPECTRA.  51 

of  displacement.    It  was  found  to  be,  between  appearance  and  evanescence 
of  fringes, 


0.119,    etc.,  cm. 

thus  scarcely  larger  than  a  millimeter,  whereas  in  the  case  where  a  grating 
(D  =  352  X  icr*  cm.)  was  used  in  place  of  P  the  range  of  displacement  was  of 
the  order  of  5  mm. 

If  the  spectra  be  regarded  with  a  prism  grating,  they  become  relatively 
long  and  short,  respectively;  but  the  phenomenon  is  none  the  less  strong, 
although  it  is  apt  to  lie  outside  of  the  two  sodium  doublets  and  not  midway 
between  them,  as  with  the  telescope.  It  seeks  out  the  line  of  coincident 
wave-lengths.  Now,  inasmuch  as  the  refraction  from  M  is  normal  and  the 
rays  virtually  retrace  their  paths  both  in  the  case  of  the  prism  and  the  grating 
(in  the  original  adjustment)  ,  it  seems  at  first  difficult  to  avoid  the  conclusion 
that  wave-trains  are  more  uniform  in  proportion  as  they  have  been  more 
highly  dispersed.  The  only  misgiving  would  be  the  fact  that  the  phenomenon 
with  prism  appears  and  vanishes  in  hair-lines,  whereas  with  gratings  it  goes 
out  rather  abruptly.  Otherwise  one  would  regard  white  light  as  consisting  of 
irregular  pulses  incapable  of  prolonged  interference,  whereas  the  dispersed 
wave  consists  of  wave-trains  in  each  color,  which  throughout  a  considerable 
number  of  wave-lengths  are  plane  polarized.  True,  if  there  is  sliding,  the 
sections  of  the  two  light-pencils,  the  points  of  which  are  capable  of  interfering 
in  pairs,  increase  in  area  proportionately  to  the  dispersion. 

Suppose  that  for  low  dispersion  the  fringes  may  be  regarded  as  extremely 
eccentric,  virtually  linear  ellipses,  the  lateral  distance  between  which  very 
rapidly  diminishes,  so  that,  since  8e  =  o.i2,  but 


2  3X10-* 

can  be  seen  by  the  given  telescope.  These  lines  would  move  behind  the  strip 
carrying  interference  fringes  asMis  displaced.  If  nowthe  dispersion  were  much 

jn 

increased,  say  from  —  =  2  X76o  for  the  prism  to  2  X  2880  for  the  grating,  the 

oX 

ellipses  would  be  much  less  eccentric  as  a  whole  and  their  lines  would  have 
grown  coarser,  so  that  many  more  would  be  visible  by  the  given  optical 
system.  As  the  dispersion  is  increased  2880/760  =  3.8  times,  the  range  of 
displacement  should  increase  similarly  to  0.12X3.8  =  4.6  cm.  The  plane 
ruled  grating  (D  =  3 52X10-*  cm.)  in  question  was  now  again  mounted  in 
place  of  P  and  under  good  illumination  the  range  0.48  cm.  was  found  experi- 
mentally. This  agrees  very  well  with  the  estimated  value.  Moreover,  on 
close  inspection  it  is  discernible  that  the  linear  phenomenon  really  consists 
of  extremely  eccentric  ellipses,  which  in  case  of  the  best  adjustments  manifest 
the  very  sharp  arrow-like  forms.  It  also  enters  and  vanishes  in  multilinear 
form,  though  the  lines  are  not  hair-lines.  Thus  the  assertion  that  increased 
uniformity  of  wave-train  accounts  for  the  long  range  of  displacement  and 


52  THE   INTERFEROMETRY  OF 

visibility  in  case  of  the  grating  is  not  warranted.  In  the  second  order  of  the 
ruled  grating  or  with  a  grating  of  higher  dispersion  (.0  =  175X10-*  cm.)  the 
field  was  too  dark  for  experiments  of  this  kind.  In  this  work,  however,  I 
obtained  the  linear  phenomenon  for  the  first  time,  from  the  double  diffraction 
of  a  film  grating. 

In  conclusion,  it  is  interesting  to  refer  to  a  relation  of  the  reversed  and  the 
inverted  spectra  and  their  interferences.  If  in  case  of  reversed  spectra  one 
of  the  superposed  pair  is  rotated  180°  in  its  own  plane,  around  an  axis  normal 
to  that  plane  and  through  the  line  of  symmetry,  the  new  pair  of  superposed 
spectra  is  an  inverted  system.  At  the  same  time  the  interferences  which  are 
ellipses  in  both  experiments  probably  rotate  their  major  axes  90°.  In  the 
case  of  reversed  spectra  this  major  axis  is  transverse,  coinciding  with  the  line 
of  symmetry  in  a  given  wave-length,  and  the  ellipses  are  extremely  eccentric, 
whereas  in  the  case  of  inverted  spectra  the  major  axis  is  probably  longitudinal. 
It  is  not  unusual  to  obtain  a  single  line  running  all  the  way  from  red  to  violet ; 
but  arrow-shaped  forms  never  occur,  so  that  the  ellipses  are  rounded  forms 
and  belong  to  distant  centers.  An  adequate  reason  for  the  highly  eccentric, 
closely  packed  elliptic  fringes  of  reversed  spectra  on  their  evolution  from  the 
round  ellipses  of  inverted  spectra  by  rotation  is  yet  to  be  given. 

23.  Apparent  lengths  of  uniform  wave-trains. — In  §  16  certain  results  were 
given  which  made  it  seem  plausible  that  the  path-differences  within  which 
interferences  are  producible  (i.e.,  the  apparent  lengths  of  uniform  wave- 
trains)  increase,  as  the  dispersion  to  which  the  incident  collimated  white 
light  is  subjected  is  made  continually  greater.  Work  with  this  quest  in  view 
is  reported  in  table  10,  the  plan  being  to  produce  the  interferences  by  one  and 
the  same  method,  but  with  a  successive  variation  of  the  dispersion  of  spectra. 
The  method,  figure  14,  was  first  selected  for  this  purpose,  inasmuch  as  the  use 
of  prisms  and  gratings  of  different  dispersive  power  at  P  meets  the  require- 
ments, while  spectra  of  the  first  and  second  order  are  equally  available. 

It  is  obvious  that  in  work  of  this  kind  the  spectra  must  be  bright,  otherwise 
the  fine  lines  will  escape  detection.  Deficient  values  will  thus  be  attained 
if  the  spectra  are  too  dark.  Moreover,  the  results  can  not  furnish  data  of 
precision,  since  the  exact  instant  at  which  fringes,  continually  decreasing  in 
size,  have  actually  vanished,  can  not  be  fixed;  and  it  is  the  fine  fringes  which 
furnish  a  considerable  amount  of  the  displacement.  The  differences,  however, 
are  so  large  that  not  only  orders  of  values  are  clearly  apparent,  but  the 
ranges  more  than  sufficiently  so  to  substantiate  the  argument. 

It  is  possible  that  the  method,  figure  14,  gives  the  half-ranges  only,  since 
the  efficient  patches  of  light,  figure  8,  can  not  cross  each  other.  The  methods 
applied  will  nevertheless  be  trustworthy,  since  they  are  identical,  the  same 
telescope  and  other  appurtenances  being  used  throughout.  Later  the  grating 
method  (fig.  3),  suitably  modified,  will  be  used.  Path-lengths  of  a  meter  or 
more  were  usually  admissible. 

In  table  10  the  first  series  of  measurements  is  obtained  with  a  60°  prism, 


REVERSED  AND   NON-REVERSED   SPECTRA. 


53 


the  dispersive  power  d6/d\  being  computed  (approximately)  from  Cauchy's 
equation,  so  that 


nearly,  and  therefore 

d8/d\  =  4B  sin  «s/2/X3  cos  (<p+8)/2 

<p  being  the  prism  angle  (60°)  and  d  the  angle  of  minimum  deviation.    The 
constant  B  was  put  4.6  Xio~u. 

TABLE  10.  —  Ranges  of  displacement,  e,  y,  for  different  dispersions.    Method  of  figure  14. 
5=4.6X10-".     it  =  1.6. 


Disposition. 

eMXio* 

ewXio8 

yXio3 
atP 

dO/d\ 

e 

Remarks. 

60°  prism  P,  90°  prism  P' 

cm. 
26 
25 
24 

cm. 

19 

18 

17 

cm. 
16 
17 
19 

760 

49°  45' 

Mean               

25 

18 

17 

Same.  P'  brought  forward 

28 
32 

29 
30 

27 
23 
26 

760 

49°  45' 

-3Q 

^O 

Ruled  gratingP,90°prismP' 

100 
100 

117 

8l 
89 

65 
76 
56 

2,880 

9°  39' 

Mean          

1  06 

Qe 

66 

Same,  mean  
Same,  second  order  from  P 
Same,  second  order  near 

161 

250 

136 
229 

108 
155 

6,030 

19°  34' 

Very  bright. 

I5° 



Film  grating  P,9O°prismP' 

173 

I84 

155 

6,400 

20°  40' 

Too  dark. 

Same  

301 

247 

183 

303 

226 

200 

Very  bright. 

Mean  

302 

236 

191 

Same,  vertical  fringes  

325 

234 

6,400 

20°  40' 

Very  bright. 

Same,  second  order  from  P 

482 
461 

422 

450 

427 

16,910 

44°  56' 

Mean 

472 

4.22 

4l8 

Same,  second  order,  ver- 
tical fringes 

EOO 

Strong  fringes. 

In  the  remaining  series  dd/d\  =  i/D  cos  d,  the  usual  expression  for  the  grat- 
ing, e  being  the  angle  of  diffraction  and  D  the  grating  space.  The  dispersive 
power  thus  increases  from  about  800  to  17,000,  over  20  times.  Throughout 
this  whole  enormous  range  good  fringes  were  obtained. 

The  values  e  show  the  displacement  of  the  opaque  mirror  M  during  the 


54 


THE   INTERFEROMETRY  OF 


presence  of  fringes,  and  of  the  opaque  mirror  N  as  specified.  Of  these,  eM  is 
systematically  larger  than  e^  for  reasons  which  do  not  appear.  The  screws 
were  of  American  and  foreign  make,  but  they  could  not  be  so  different.  It 
is  due  very  probably  to  residual  curvature  in  the  mirrors  and  surfaces,  whereby 
fringes  on  the  left  (AT)  vanish  sooner  than  those  on  the  right  (M) .  The  datum 
y  is  the  displacement  of  the  right-angled  reflecting  prism  Pf,  parallel  to  the 
component  rays  bb'.  This  value  is  smaller  than  e  for  reasons  already  discussed 
in  §  16.  All  measurements  were  frequently  repeated  and  the  means  finally 
taken  for  comparison  with  dQ/dK. 

In  the  third  series  (ruled  grating  and  concave  grating)  with  specially  brilliant 
spectra,  the  phenomenon  of  figure  32  was  observed.  A  wide  field  of  faint 
fringes  was  visible,  enormously  accentuated  and  clear  in  the  narrow  strip  of 
the  linear  phenomenon.  As  the  micrometer  mirror  at  M  moves  forward,  these 
faint  fringes  shift  bodily  across  the  stationary  bright  linear  strip,  beginning 
therefore  with  the  pattern  a  and  ending  with  b.  The  faint  fringes  follow  the 
rules  of  displacement  interferometry. 

6 


/- 
o  /a 


33 


32 


4-\          34 


0      4-      8      ft     16     20 

In  addition  to  the  data  of  the  table,  a  large  number  of  miscellaneous  tests 
were  made  with  the  reflecting  prism  in  different  positions.  Unless  brought 
too  far  to  the  rear,  when  the  beams  are  lost  at  the  edge  and  e  too  small,  the 
results  for  fine  and  coarser  fringes  were  of  the  same  order. 

The  values  of  y  have  been  graphically  given  in  figure  33 ;  those  for  e  are 
not  sufficiently  regular  in  the  dispersive  powers  above  i  ,000  for  this  treament . 
It  is  probable,  for  instance,  that  at  16,900  the  sliding  along  the  prism  surface 
is  interfered  with.  All  the  data,  in  consideration  of  their  limitations,  bear  out 
the  inference  that  the  range  of  displacement  within  which  fringes  are  seen 
increases  in  marked  degree  with  the  dispersion.  The  average  initial  ratio 
2\/(d6/d\)  is  about  60X10-*  cm. 

A  very  surprising  result  in  these  experiments  is  the  efficiency  of  the  film 
grating  in  series  IV  and  V,  not  only  in  the  first  but  in  the  second  order  of 
spectra. 


REVERSED  AND   NON-REVERSED   SPECTRA. 


55 


After  these  experiments  an  attempt  was  made  to  obtain  similar  results  with 
the  more  comprehensive  method  of  two  gratings  (transmitting  G  and  reflecting 
£')  of  figure  3,  above.  But  here  the  choice  of  gratings  with  appropriate  con- 
stants was  limited  and  with  high  double  dispersion  the  fields  were  apt  to  be 
too  dark.  Good  results  were  obtained  with  the  60°  prism  and  concave  grating 
and  with  the  ruled  grating  together  with  the  latter.  In  the  method  of  figure  3 , 
the  second  angle  of  diffraction  is  necessarily  greater  than  the  first,  tf>  9.  To 
obviate  this  difficulty  the  method  was  modified  for  prisms  as  in  figure  20, 
where  G  is  the  concave  grating,  T  a  strong  lens  near  its  focus,  and  m  and  n 
auxiliary  mirrors.  If  this  method  is  used  for  highly  dispersive  gratings  (G 
replacing  P),  the  rays  must  be  crossed  as  shown  in  figure  34.  The  fringes 
were  found  in  this  case  when  G  was  a  film  grating,  but  the  work  had  to  be 
abandoned,  as  the  spectra  were  dull. 

The  data  given  in  table  n  again  show  marked  increase  of  displacement 
with  the  dispersion  dB/dK,  though  it  is  not  proportional  here.  The  method 
with  two  gratings  lacks  the  brilliancy  of  the  prism  method. 

TABLE  n. — Range  of  displacement  for  different  dispersive  powers.    Method  of  figure  20, 

full  displacement. 


Details. 

•m 

•» 

de 
d\ 

e 

Remarks. 

60°  prism  and  concave  grating  
Ruled  and  concave  grating 

cm. 
0-357 
.300 

514 

cm. 

0.315 
.282 

C2O 

760 
760 

2  880 

49°  45' 
49  45 
9  39 

Bright. 
Too  dark. 
Bright  arrows. 

Film  and  concave  grating  

•509 

2,880 
6,400 

9  39 

20    40 

Dark. 
Too  dark. 

24.  Normal  displacement  of  mirrors  (5  =  o). — This  desideratum  was  secured 
in  the  original  methods,  in  which  a  single  grating  was  used  for  both  diffractions. 
Rays  in  such  a  case  have  to  cross  the  grating  somewhat  obliquely  to  the  hori- 
zontal. The  method,  furthermore,  is  restricted  to  the  linear  vertical  fringes, 
which  are  not  useful  if  practical  measurement  is  aimed  at. 

In  the  methods  with  a  right-angled  reflecting  prism  (fig.  14),  this  result  is 
easily  secured  by  displacing  the  prism.  In  all  other  methods  (5>o),  the  dis- 
placement of  mirror  is  accompanied  by  sliding  of  the  pencil  along  it.  The 
effect,  as  has  been  shown,  is  not  negligible.  It  therefore  seemed  desirable  to 
devise  other  methods  in  which  5  =  o,  and  figure  35  is  a  device  of  this  kind. 

Here  G  and  G'  are  two  identical  gratings,  the  first,  G,  receiving  the  light  L 
from  a  collimator.  The  component  pencils  a,  a'  pass  through  the  half-silvered 
plate  H,  and  thence  (b,  b')  to  the  opaque  mirrors  M  and  N,  one  or  both  on  a 
micrometer.  The  pencils  6  and  b',  impinging  normally,  retrace  their  paths 
and  are  thereafter  reflected  at  the  plate  H  into  c  and  c' .  These  strike  the 
second  grating  G'  at  the  proper  angle  of  diffraction  and  thereafter  enter  the 
telescope  T  together.  The  path  of  rays  is  symmetrical  throughout.  White 
light  is  screened  off  at  d.  Reversed  spectra  are  seen  at  T.  Unfortunately  the 
only  identical  gratings  at  my  disposal  were  film  gratings  of  high  dispersion 


56  THE  INTERFEROMETRY   OF 

(D  =  162  X  io-*) .  As  a  result  of  the  two  diffractions  and  the  half -silver  reflec- 
tion, the  spectra  were  too  dull  to  make  it  worth  while  to  look  for  fringes,  at 
length.  None  was  discernible  after  some  searching,  to  my  regret,  for  the 
method  itself  has  many  points  of  interest  and  with  gratings  of  low  dispersion 
it  succeeds. 

Later  I  stripped  a  celluloid  film  from  the  ruled  grating  (D  =  352  X  icr6)  and 
mounted  it  by  simple  stretching.  Using  the  original  grating  at  G  and  the  film 
at  G',  the  fringes  were  found  with  some  patience.  The  spectra  were  fairly 
bright  (arc  lamp)  and  the  fringes  reasonably  strong;  but  they  admitted  of  a 
displacement  of  only  i .8  mm.,  in  spite  of  the  vanishing  angle  6  =  0.  Possibly 
this  small  displacement  is  due  to  the  imperfect  film  grating. 

In  further  experiments  I  half -silvered  plates  of  glass  to  different  density. 
In  this  work  I  obtained  adequately  bright  spectra  and  practically  perfect 
fringes,  but  the  range  of  displacement  (3  =  o)  could  not  be  increased  above 
1.8  mm.  Within  this  interval  the  fringes  seem  to  change  form  but  little, 
thinning  only  being  evident.  Then  they  become  dull  and  vanish. 

The  method  was  temporarily  modified,  as  shown  in  figure  36,  where  G  is  a 
ruled  grating  (D  =  3 52X10-*  cm.),  G'  a  film  grating  (D  =  176X10-*  cm.),  and 
m  an  opaque  mirror.  This  naturally  introduces  an  angle  8=  6'—  0,  which  is 
negative  when  6'>  6,  or  the  grating  G  shows  greater  dispersion.  The  mirror 
is  limited  in  breadth,  so  that  the  rays  a,  a'  have  free  access  to  M  and  N.  The 
fringes  were  found  after  some  trouble,  for  the  transmitting  grating  G'  also 
acts  as  a  reflecting  grating  if  the  rays  b,  b'  fall  upon  it,  and  it  is  not  always 
easy  to  separate  these  two  cases,  each  of  which  will  give  fringes  on  proper 
adjustment.  The  spectra  are  very  bright  and  the  range  was  about  2  mm. 

The  same  method  was  now  carried  out  with  prisms,  as  shown  in  figure  37, 
where  L  is  the  incident  pencil  from  a  collimator,  P  and  P'  right-angled  prisms, 
I  the  half-silvered  plate,  N  and  M  opaque  mirrors  on  micrometers,  T  the 
telescope.  The  spectra  seen  at  T  have  each  been  four  times  refracted  and 
twice  reflected,  at  M  and  H.  They  are  very  bright,  so  that  a  very  fine  slit 
(here  specially  desirable)  is  available.  The  sodium  lines  are  not  separated. 
On  bringing  the  spectra  to  overlap  at  their  corresponding  edges,  the  fringes 
were  found.  They  are  peculiar,  inasmuch  as  they  show  the  phenomenon  of 
figure  32,  but  with  the  faint  fringes  curved  and  more  prominent  than  hereto- 
fore. In  other  words,  the  faint  phenomenon  shifts  across  the  field  from  side 
to  side,  but  is  enormously  accentuated  at  the  transverse  strip  of  the  linear 
phenomenon.  Narrowing  the  incident  pencil  broadens  and  blackens  the 
fringes.  They  may  be  obtained  in  the  gap  between  two  spectra  just  separated. 
The  range  of  displacement  within  which  fringes  are  seen  was,  however,  very 
small,  not  exceeding  0.2  mm.  This  is  a  characteristic  of  these  fringes  and  in 
keeping  with  the  low  dispersion. 

It  is  interesting  to  note  that  the  systems  figures  35,  36,  37  constitute  an 
element  of  a  direct-vision  spectroscope.  It  has  been  shown  that  it  can  be 
made  quite  powerful. 

The  same  method,  figure  37,  was  now  used  with  two  60°  prisms  of  highly 


REVERSED  AND   NON-REVERSED   SPECTRA. 


57 


refracting  glass.  In  contrast  with  the  ease  in  which  the  adjustments  were 
made  with  the  symmetrical  45°  prisms,  the  corresponding  work  with  60°  prisms 
proved  to  be  exceedingly  difficult.  True,  the  prisms  were  large,  including 
long  glass-paths.  The  sodium  lines,  now  clearly  separated,  were  markedly 
curved,  so  that  on  placing  them  near  together  they  either  assumed  an  O-shape 
or  an  X-shape.  But  the  spectra  were  brilliant  and  nothing  appeared  to  mili- 
tate against  a  successful  result.  But  it  was  not  until  after  days  of  searching 
that  the  fringes  were  found,  and  then  only  with  two  prisms  of  different,  highly 
refracting  glass.  They  were  not  quite  uniform,  but  it  seemed  impossible  to 
improve  them.  The  ranges  of  displacement  were  found  to  be  0.088  to  0.093 
cm.  with  the  electric  arc,  0.103  cm.  with  (condensed)  sunlight.  This  is  again 
in  accord  with  the  large  increase  of  dispersion,  the  range  of  displacement  being 
about  five  times  greater  than  was  the  case  with  45°  prism  of  less  refracting 
glass.  - 


35 


25.  Diffraction  at  M,  N,  replacing  reflection. — The  present  method  of 
observing  interferences  in  the  zero,  first,  second,  third,  and  even  fourth 
order,  successively,  without  essential  change  of  the  parts  of  the  apparatus,  is 
noteworthy.  I  happened  to  possess  a  plane  reflecting  grating  (D  X  lo"6  =  200) , 
cut  into  two  equal  parts  by  a  section  parallel  to  the  rulings,  and  it  was  there- 
fore easy  to  devise  the  method.  In  figure 
38,  the  incident  light  L  from  the  colli- 
mator  is  separated  into  two  component 
beams  a  and  a'  by  the  60°  prism  P.  This 
is  essential  here,  as  an  abundance  of  light 
is  needed  (sunlight  should  be  focused  by 
a  large  lens  of  long  focus  (5  feet)  on  the 
slit).  The  rays  a,  a'  are  then  either 
reflected  or  diffracted  in  any  order  by  the 
plane  reflecting  gratings  G,  G'  into  the  collinear  rays  b,  b'.  They  are  then 
reflected  by  the  silvered  right-angled  prism  P'  and  observed  in  a  telescope 
at  T.  G  and  G'  and  if  possible  also  P'  should  be  on  micrometers,  so  that 


58 


THE   INTERFEROMETRY  OF 


corresponding  displacements  e,  e',  normal  to  G  and  G'  and  y  in  the  direction 
bb',  may  be  registered. 

The  adjustments, if  symmetry  were  demanded,  would  be  cumbersome;  for, 
in  addition  to  precise  modification  of  the  position  and  orientation  of  the  prisms 
P,  P',  the  grating  requires  fine  adjustment  and  a  means  of  securing  parallelism 
of  the  rulings.  But  an  approximate  adjustment  does  very  well  and  no  pains 
were  taken  in  the  first  experiments  to  secure  symmetry.  The  spectra  were 
intensely  brilliant  in  the  low-order  work;  but  even  in  the  fourth  order  the 
light  was  adequate.  One  may  note  that  here  the  gratings  do  not  reverse  the 
dispersion  of  the  prism  P,  though  this  is  relatively  small.  Table  12  is  an 
example  of  results : 

TABLE  12. — Ranges  of  displacement  varying  with  dispersion.    Paired 
gratings  and  60°  prism.    0=46°.    8=44°.    x=2ecosS/2. 


Order 

Observed 
eXio3 

*Xio3 

i 

d$/d\ 

Remarks. 

cm. 

cm. 

o 

38 

70 

22° 

760 

1  80 

200 

180 

35i 

100 
12.8° 
31-2° 

}  3.490 

200 

351 

2 

420 

420 

777 

1              -«.<> 

^ 

400 
380 

I       721 

L       3-5 
J     40-5° 

}   6,440 

3 

320 
300 
520 
520 

|     573 
|     962 

-6.4° 
50.4° 

}   9,930 

/Fringes  lost  at 
\      edge? 

4 

580 
580 
520 

1   1070 

1-17-6° 
61.6° 

}  14.800 

("Fringes  faint. 
[Fringes  faint. 

450 

| 

The  fringes  in  the  zero  order  were  good  and  strong,  not  inferior  to  any  of 
the  others,  but  unfortunately  too  short-lived.  In  the  fourth  order  the  fringes 
are  weak  (although  the  enormous  sodium  doublets  stand  out  clearly),  doubt- 
less from  excess  of  extraneous  light.  Here  also  it  is  difficult  to  prevent  the 
beam  from  vanishing  at  the  edge  of  the  prism  P'.  Hence  the  anomalously 
small  displacement,  a  discrepancy  which  is  already  quite  manifest  in  the 
third  order. 

The  present  experiments  furnish  a  striking  example  of  the  uniform  breadth 
of  the  strip  of  spectrum  carrying  the  fringes,  quite  apart  from  the  dispersion 
of  the  spectra.  In  the  prism  spectrum,  where  the  sodium  doublets  are  indi- 
cated by  a  hair-line  just  visible,  to  the  fourth-order  spectra,  where  they  stand 
apart  like  ropes,  the  linear  phenomenon  has  the  same  width. 

The  computation  of  the  dispersive  power  in  these  cases  is  peculiar.  It  will 
be  seen  from  figure  38  that  the  angle  8  =  44°  between  the  incident  ray  a  and 


REVERSED   AND   NON-REVERSED   SPECTRA. 


59 


the  diffracted  ray  b  is  constant  and  is  8=6-\-i  in  the  first  and  second  and 
8  =  6—  i  in  the  third  and  fourth  orders.    Hence  in  succession 


sin  (5  —  i)  —  sin  i  =  X/Z) 
sin 


sin  (5 — i)  —  sin  i  =  z\/D 
sin(3-H)-r-sint  = 


from  which  equations  the  angle  i  may  be  computed.    I  did  this  with  sufficient 
accuracy  graphically,  and  the  values  of  *  and  6  so  found  are  given  in  table  12. 
Since  dQ—di,  apart  from  sign,  it  follows  that  the  dispersing  power  is 

-de/d\=n/D  (cost+cos  («—»")) 

where  n  is  the  order  of  the  spectrum  and  i  changes  sign  in  the  third  and  fourth 
orders.  With  the  values  of  i  given,  the  data  for  d 0/dK  in  the  table  were  finally 
computed.  The  dispersive  power  of  the  prism  was  computed  as  above  and 
is  to  be  added  to  all  the  succeeding  dispersive  powers.  Figure  39  shows  the 
relation  between  the  dispersive  pow- 
ers and  the  path-difference  x  =  ze 
cos  5/2  computed  from  the  observed 
range  of  displacement  e  of  the  grat- 
ing G.  The  largest  values  of  x  are 
taken,  as  they  are  the  most  proba- 
ble. The  effect  of  dispersion  here 
breaks  down  in  the  third  and  fourth 
orders,  as  already  stated,  probably 
from  incidental  causes.  For  the 


rfftjfci 


39 


10 


spectra  themselves  were  still  ade- 
quately bright,  but  the  fringes  were 
faint  for  some  reason  and  I  failed  to  make  them  stronger.  The  rate  x/(dB/dK) 
is  here  about  120X10-*  initially.  This  is  larger  than  above,  owing  to  the 
differences  of  apparatus  used,  etc. 

26.  Experiments  with  the  concave  grating. — As  there  was  an  excellent 
Rowland  grating  in  the  laboratory  with  a  6-foot  radius  and  a  grating  space 
D=i77Xicr*  cm.,  it  seemed  worth  while  to  obtain  the  interferences  with  it. 
I  had  hoped  to  doubly  diffract  the  rays  at  the  same  grating,  but  there  is  not 
light  enough  to  make  this  method  fruitful.  Accordingly  the  device,  figure  40, 
promised  the  best  results,  where  L  is  a  convergent  pencil  of  sunlight,  S  the 
slit.  The  pencil  L  is  carried  above  the  grating  G  to  the  opaque  mirror  m, 
whence  it  is  reflected  to  the  grating  and  diffracted  into  the  component  beams 
a  and  a'.  These  are  in  turn  reflected  by  the  opaque  mirrors  M  and  N  (on 
micrometers)  into  the  pencils  b  and  b',  nearly  collinear.  The  latter  are  then 
reflected  by  the  silvered  right-angled  prism  P  to  the  common  focus  F,  to  be 
observed  with  the  lens  T.  The  spectra  are  very  bright  and  highly  dispersed 
and  are  easily  made  to  overlap  on  their  contiguous  edges  (parallel  to  a  Fraun- 
hofer  line  D,  for  instance)  sufficiently  to  show  the  linear  phenomenon  of 
reversed  spectra. 


60 


THE   INTERFEROMETRY   OF 


The  fringes  are  easily  found  and  splendid,  though  they  are  not  superior  in 
this  respect  to  the  fringes  obtained  by  other  methods.  P  is  also  on  a  microm- 
eter with  the  screw  in  the  mean  direction  b  b'.  The  range  of  displacement  of 
P  was  about  y  =  0.5 5  cm.,  so  that  the  available  path-difference  is  considerably 
above  a  centimeter.  Within  this  the  fringes  pass  from  fine  hair-lines  through 
a  maximum  and  back  again,  apparently  without  rotation.  To  enlarge  the 
fringes  the  grating  G  may  be  tilted  in  its  own  plane  or  a  similar  adjustment 
made  at  P. 

The  longitudinal  axes  of  the  spectra  (wire  across  the  slit)  are  not  in  focus 
with  the  D  line;  but  though  hazy  they  suffice  for  adjustment.  Naturally  the 
distance  FbaG  is  the  radius  of  the  grating  R  (6  feet)  and  the  distance  Sm-\-mG 
is  R  cos  6.  The  distances  are  approximately  laid  off  and  the  observer  at  T 
may  then  push  the  mirror  M  fore  and  aft  by  the  aid  of  a  lever  till  the  Fraun- 
hofer  lines  are  sharp. 

Unfortunately  the  spectra  as  a  whole  are  shifted  by  the  micrometers,  to- 
gether when  P  moves  and  separately  when  M  or  N  moves. 

27.  Polarization. — The  two  rays  obtained  from  calc  spar,  if  corrected  for 
polarization,  should  be  available  for  interferences  of  the  present  kind.  Nat- 
urally an  achromatized  calc-spar  prism  (C,  fig.  41)  is  most  convenient  for  the 


40 


41 


purpose.  White  light  from  a  collimator,  L,  is  doubly  refracted  by  this  prism 
C,  and  the  extraordinary  ray  a,  just  missing  the  grating  G  (above  or  on  the 
sides),  impinges  on  the  opaque  mirror  M  and  is  thence  reflected  to  the  grating 
G.  The  ordinary  ray  a'  is  reflected  from  the  opaque  mirror  N  and  thence  also 
reaches  the  grating.  These  two  pencils,  b,  b',  are  directly  reflected  by  the 
grating  (£>Xio6  =  2Oo  cm.)  into  Rm  and  Rn  and  diffracted  into  £>'mand  £>'.. 
By  suitably  adjusting  the  grating  between  C  and  M  and  inclining  it  as  shown, 
two  coincident  spectra  may  be  made  to  pass  along  the  common  direction  c 
to  the  telescope  at  T.  These  spectra  are  quite  intense.  One  is  somewhat 


REVERSED   AND   NON-REVERSED   SPECTRA.  61 

more  dispersed  than  the  other,  owing  to  the  difference  of  angles  of  incidence 
i,  i'  and  to  the  residual  or  differential  dispersion  at  the  calc-spar  prism,  for 
only  the  a  pencil  has  been  adequately  achromatized.  In  my  apparatus  the 
distances  CM  and  GM  were  about  130  and  too  cm.  and  the  distance  NM 
roughly  30  cm.  N  is  on  a  micrometer,  as  it  is  nearer  to  the  observer  at  T, 
though  it  would  be  preferable  to  put  M  on  a  micrometer. 

To  obtain  interferences  of  the  present  kind,  the  plane  of  polarization  of 
either  a  or  a'  must  be  rotated  90°.  This  may  be  done  by  two  quarter-wave- 
length micas  M,  in  a',  the  first  of  which  (set  at  45°  as  usual)  produces  circu- 
larly polarized  light  and  the  second  then  erects  the  vibration  into  parallelism 
with  the  vibration  of  the  pencil  a.  Other  methods  will  presently  be  resorted  to. 

With  this  adjustment  fringes  were  found  at  T  after  some  searching.  They 
were  large,  but  occurred  in  a  transverse  strip  of  spectrum  about  A  #2/2  in 
width.  .True,  the  spectra  are  without  reversion;  but,  as  stated  before,  one 
was  about  one-fifth  longer  than  the  other.  This  is  probably  the  reason  for 
the  narrowness  of  the  strip,  for  the  fringes  should  otherwise  fill  the  spectrum. 

The  fringes  as  first  found  admitted  a  displacement  of  N  of  about  0.3  cm. 
only.  They  were  hard  to  control,  needed  sharp  longitudinal  adjustment,  and 
when  lost  were  difficult  to  find  again.  They  rotate  from  and  to  vertical  hair- 
lines, through  a  horizontal  maximum.  They  are  always  found  in  the  line  of 
symmetry,  which  for  unequal  spectra  moves  more  rapidly  than  the  Fraun- 
hofer  lines  if  they  are  separated.  The  nearer  quarter-undulation  plate  at 
W  may  be  considerably  rotated  without  quite  destroying  the  fringes. 

If  the  double  quarter-wave-length  plate  is  suitably  put  in  the  a  beam, 
results  of  the  same  kind  are  naturally  obtained.  If  a  single  quarter- wave- 
length plate  is  placed  in  each  beam,  at  45°,  both  will  be  circularly  polarized, 
the  position  of  the  micrometer  being  intermediate.  I  tested  this  case  at  some 
length,  but  found  no  interferences,  as  was  to  be  expected.  Circularly  polar- 
ized rays  of  the  same  sense  do  not  interfere. 

In  place  of  the  achromatized  calc-spar  prism,  a  double-image  Wollaston 
prism  or  a  double-image  Fresnel  prism  (rotary  polarization)  could  be  used. 
Unfortunately  I  had  neither  of  these,  but  the  latter  in  connection  with  an 
analyzing  nicol  would  have  been  worth  testing. 


CHAPTER  II. 


THE  INTERFERENCES  OF  INVERTED  SPECTRA. 

28.  Introductory. — If  two  identical  spectra  are  superposed  in  such  a  way 
that  one  is  rotated  180°,  on  a  transverse  axis  (parallel  to  the  Fraunhofer 
lines),  with  respect  to  the  other,  it  will  be  convenient  to  refer  to  the  phenomena 
resulting  and  described  in  Chapter  I  as  the  interferences  of  reversed  spectra. 
On  the  other  hand,  if  one  of  the  superposed  spectra  has  been  rotated  180° 
with  respect  to  the  other  on  a  longitudinal  axis  (parallel  to  the  length  of  the 
spectrum),  we  may  refer  to  the  interferences  as  those  of  inverted  spectra. 
The  absence  of  either  of  these  adjustments  would  then  be  accentuated  as 
non-reversed  or  non-inverted. 

In  the  case  of  inverted  spectra,  therefore,  we  are  dealing  with  phenomena 
of  virtually  homogeneous  light,  exhibited  throughout  the  length  of  the  spec- 
trum. Such  experiments  were  made  cursorily  in  my  first  paper  on  the  sub- 
ject,* but  the  phenomenon  is  very  peculiar,  apparently  anomalous,  and  further 
treatment  is  therefore  desirable. 

29.  Apparatus.    Non=inverted  spectra. — The  apparatus  formerly  used  for 
long  optical  paths  is  difficult  to  manipulate.     It  has  therefore  been  simplified 
in  the  present  method  and  used  for  short  distances,  so  as  to  be  wholly  in  the 
observer's  control.    The  parts  of  the  apparatus  are  conveniently  assembled  as 
in  a  preceding  experiment,  except  that  the  slit  which  furnishes  the  collimated 
light  L,  figure  42,  is  now  horizontal — i.e.,  at  right  angles  to  the  edge  of  the 
sharp  prism  P  (about  20°  at  apex).    The  rays  a  and  a'  (horizontal  blades), 
after  leaving  the  opaque  mirrors  M  and  N,  are  then  reflected  from  the  sides 
of  the  right-angled  prism  P'  into  c  and  c' .    Thus  far  the  light  is  white;  but 
c  and  c'  are  now  diffracted  by  the  grating  G  with  its  rulings  horizontal  (par- 
allel to  slit)  and  with  the  interposition  of  an  auxiliary  prism  p  (edge  hori- 
zontal), the  two  spectra  due  to  c  and  c'  are  observed  by  a  telescope  at  T. 

The  experiment  succeeds  best  with  sunlight.  The  triangle  of  rays,  a,  a', 
br,  b,  is  first  made  isosceles  and  horizontal  by  adjusting  P,P',  being  set  midway 
and  symmetrically  between  M  and  N.  The  two  spectra  in  the  telescope 
(vertical  ribands)  are  now  made  to  overlap  at  their  inner  edges  and  the  two 
sodium  doublets  placed  accurately  in  contact,  by  aid  of  the  adjustment 
screws  on  the  mirrors  M  and  N.  The  latter  are  then  to  be  moved  micromet- 
rically  (Fraunhofer  slide)  until  the  fringes  appear.  The  experiment  is  not 
an  easy  one. 

It  is  obvious  from  figure  42  that  the  two  superimposed  spectra  are  non- 

*  Am.  Journal,  XL,  pp.  486  to  498,  1915,  §  4;  Carnegie  Inst.  Wash.  Pub.  249,  Chap.  I. 
62 


REVERSED  AND   NON-REVERSED   SPECTRA. 


63 


inverted  or  direct.  Hence  with  a  wide  slit,  or  insufficiently  homogeneous 
light  in  the  direction  of  a  given  Fraunhofer  line,  no  fringes  will  appear.  Such 
fringes  as  may  in  any  case  be  found  will  not,  therefore,  much  outlast  the 
Fraunhofer  lines,  so  far  as  width  of  slit  is  concerned.  Furthermore,  even  if 
the  slit  is  narrow  and  the  spectra  coordinated,  there  will  be  no  fringes  obtain- 
able if  the  superimposed  solar  spectra  are  quite  unbroken  —  i.e.,  without  in- 
cidental furrows  in  the  direction  of  their  length  (normal  to  the  Fraunhofer 
lines).  For  it  is  to  be  noticed  that  the  slit  is  horizontal,  and  therefore  there 
is  no  observable  diffraction  —  i.e.,  virtually  no  slit  in  the  horizontal  direction. 
If,  however,  the  spectrum  field  is  interrupted  longitudinally  by  a  thin 
(o.i  mm.)  wire  drawn  across  the  slit  (or,  much  better,  by  very  fine  specks  of 
dust  lying  incidentally  within  the  slit),  then  these  fine  opaque  objects  will 
effectively  replace  the  slit,  or  act  analogously  to  a  slit  in  the  horizontal  di- 
rection. Hence  fringes  will  appear  when  the  path-difference  is  sufficiently 
small,  associated  with  the  geometric  shadow  of  the  opaque  objects,  through- 
out the  length  of  the  spectrum. 


42 


We  have  here,  therefore,  a  peculiar  case  of  the  diffraction  of  a  rod,  from 
two  separately  controlled  half -wave-fronts.  The  fringes  at  one  extreme  of 
adjustment  of  mirror  M  begin  with  fine  horizontal  lines,  which  on  moving  M 
incline  and  enlarge  until  they  gain  the  maximum  of  size  in  the  vertical  direc- 
tion. After  this,  on  further  motion  of  M  in  the  same  direction,  they  incline 
further,  diminish  in  size,  and  finally  become  horizontal  and  hair-like  again. 
They  move  along  the  horizontal  axis  with  M,  subject  to  the  equation 

de  __        X 
dn       2cos  d/2 

where  5/2  is  the  angle  of  incidence  at  M,  X  the  wave-length,  and  de/dn  the 
normal  displacement  of  M  per  fringe. 

Within  the  region  of  overlapping  spectra  each  longitudinal  black  line  is 
covered  from  end  to  end  with  fringes,  the  strip  being  about  of  AA  width 
or  more  if  the  line  is  thinner.  A  broad  black  line  (thicker  wire  across  slit) 


64 


THE   INTERFEROMETRY  OF 


shows  the  fringes  at  its  edges  only.  While  the  spectrum  may  thus  be  covered 
with  independent  strands  of  fringes,  they  all  move  or  change  phase  together. 
The  range  of  displacement  of  the  micrometer-screw  is  about  2  mm.  When 
the  spectra  are  about  to  separate — i.e.,  when  overlapping  ceases — the  fringes 
are  apt  to  be  particularly  wide,  say  even  4AA,  and  they  may  be  seen  in 
the  gap  between  spectra.  In  general,  the  degree  of  coarseness  of  fringes  de- 
pends on  the  adjustment  for  parallelism  of  spectra. 

30.  Apparatus  and  results  for  inverted  spectra. — The  preceding  apparatus 
may  be  modified  as  shown  in  figure  43 .  Here  L  is  a  collimated  beam  of  white 
sunlight  from  the  vertical  slit,  G  a  transmitting  grating.  The  component 
pencils  a  and  a'  of  spectrum  light  impinge  on  the  opaque  mirrors  M  and  N, 
and  are  then  reflected  in  b  and  b'  to 
reach  the  silvered  sides  of  the  right- 
angled  prism  P'.  This  is  now  placed 
with  its  sides  at  45°  to  the  horizontal 
and  its  edge  in  the  direction  of  the 
incident  beam  L  prolonged.  Hence 
the  two  pencils  b  and  b'  are  reflected 
vertically  upward  (fig.  44)  and  appear 
as  two  identical  and  parallel  spectra 
in  the  field  of  a  telescope,  vertically 
above  P'.  Observation  is  conveniently  made  downward  to  obviate  additional 
reflection.  If  the  triangle  a,  a',  b',  b  is  horizontal  and  isosceles  there  is  no 
difficulty.  On  slightly  moving  the  adjustment  screen  on  M,  N,  P'  (which 
must  be  revolvable  on  a  vertical  axis,  and  P',  or  the  beam  L,  movable  up  or 
down),  to  bring  the  two  spectra  into  coincidence  along  a  given  longitudinal 
axis,  and  a  transverse  axis  like  the  D  lines,  the  two  spectra  are  symmetri- 
cal— i.e.,  mirror  images  one  of  another  with  respect  to  the  longitudinal  axis. 
If,  now,  the  mirror  M  is  moved  on  a  micrometer,  the  fringes  of  inverted  spec- 
tra appear  when  the  path-differences  are  nearly  the  same. 


43 


a       b      c       d      e, 

Obtained  at  short  distances  (of  the  order  of  a  foot)  on  the  cast-iron  block 
(Chap.  I,  fig.  3),  these  fringes  are  quiet  and  better  circumstanced  for  obser- 
vation; but  their  characteristics  are  the  same  as  found  for  them  before  (Chap. 
I,  §  4,  Carnegie  Inst.  Wash.  Pub.  249).  They  lie  within  a  narrow  strip  at 
the  line  of  symmetry  of  the  two  superposed  spectra,  running  from  end  to 
end  and  in  breadth  about  three  times  the  distance  apart  of  the  sodium  lines 
(sAA).  When  the  mirror  M  is  moved  micrometrically  in  one  direction, 
the  fringes  begin  to  appear  as  fine  hair-lines  (a,  fig.  45)  parallel  to  the  D  lines. 
These  fine  fringes  gradually  coarsen  and  rotate  until  they  reach  their  max- 


REVERSED  AND   NON-REVERSED   SPECTRA.  65 

imum  size  c,  when  they  are  perpendicular  to  the  D  lines.  A  single  black  or 
bright  line  may  here  extend  from  end  to  end  of  the  spectrum.  Thereafter 
they  grow  finer  (rotating  again)  in  the  same  way  until  they  vanish,  e,  as 
they  began.  The  total  angle  of  rotation  is  thus  180°. 

If  the  two  beams  b  and  b',  figure  43,  are  moved  nearer  the  edge  of  the 
prism,  the  fringes  become  larger,  but  usually  not  much.  At  least  I  did  not,  at 
first,  succeed  in  separating  the  fringes  much  beyond  the  DiDz  distance.  The 
width  of  the  longitudinal  interference  strip  remains  unchanged.  If  the  light 
is  removed  at  the  line  of  symmetry  (wire  across  slit),  the  fringes  are  sharply 
outlined  across  the  black  line.  Being  so  small,  they  are  naturally  always 
sharp  and  vivid.  The  range  of  displacement  of  M  within  which  fringes  are 
visible  was  about  0.2  cm.  for  the  given  grating  (D  =  352  X  iQ"6),  corresponding 
to  the  complete  rotation  180°,  instanced  above.  If  the  slit  is  widened,  the 
fringes  slightly  outlast  the  Fraunhofer  lines.  They  also  lie  in  the  principal 
focal  plane. 

The  remarkable  feature  of  these  fringes  is  the  definite  breadth  of  strip, 
from  red  to  violet,  within  which  they  lie.  With  full  wave-fronts  the  stria- 
tions  look  as  though  they  were  cut  between  two  parallel  lines,  3AA  apart, 
In  some  adjustments,  however,  suggestions  of  concealed  fringes,  very  faint 
prolongations  of  the  strongly  marked  striations,  are  unmistakable. 

31.  Wave=fronts  narrowed. — The  longitudinal  strip  within  which  the 
interferences  lie  is  very  sharply  limited  in  breadth,  as  has  been  stated.  It 
may,  however,  be  broadened  by  screening  off  the  white  beam,  as  it  leaves  the 
collimator,  from  below.  When  the  whole  length  of  slit  is  utilized,  the  strip 
may  not  be  more  than  AA  in  width.  As  the  vertical  blade  or  beam  of  white 
light  is  cut  off,  more  and  more  from  below,  the  strip  increases  to  a  width  of 
4AA,  and  when  the  two  inverted  spectra  in  the  field  of  the  telescope  begin 
to  separate  at  the  line  of  symmetry,  the  strip  may  be  over  ictDiD^  in  width. 
It  is  obvious  that  in  such  a  case  the  light  comes  from  near  the  horizontal  top 
edge  of  the  prism.  The  wave-fronts  are  slit-like.  The  field  within  which 
diffraction  is  perceptible  in  the  telescope  increases  in  breadth  as  the  colored 
wave-front,  incident  at  the  objective  of  the  telescope  and  parallel  to  the 
spectrum,  decreases  in  breadth  in  the  same  direction.  In  figure  46,  a  and  b 
are  the  two  component  beams  from  the  two  faces  of  the  prism,  respectively. 
The  focus  is  at  /.  The  arrows  show  the  effect  of  narrowing.  The  oblique 
rays  (omitted)  are  similarly  affected  and  in  step  with  the  axial  rays  shown. 
But  the  fringes  are  not  changed  in  size.  They  may,  however,  be  definitely 
changed  in  inclination.  Size  results  from  the  anterior  relations  of  the  spectra 
(distance  between  paired  pencils),  and  not  from  the  width  of  wave-front. 

These  inverted  fringes  admit  of  much  magnification.  With  a  strong  tele- 
scope (magnification  about  15)  they  are  quite  sharp  only  in  a  part  of  the 
magnified  spectrum  and  grow  vague  beyond,  showing  that  the  component 
spectra  are  not  quite  identical  after  the  two  reflections.  When  not  quite  in 
adjustment,  the  strip  is  liable  to  exhibit  separate  oblique  strands,  lying  within 


66  THE   INTERFEROMETRY  OF 

the  same  strip  or  region  (fig.  47)-  They  may  become  sharp  by  changing 
the  focal  plane  of  the  eye-piece.  When  parallel  to  the  length  of  the  spectrum 
and  seen  in  a  strong  telescope,  about  7  lines  alternately  black  and  white  may 
be  counted.  The  whole  lie  within  a  strip  of  not  much  above  AA  width. 
The  range  of  displacement  of  M  for  a  rotation  of  180°  of  fringes  is  about 
0.25  cm.  Change  of  size  of  fringes  from  red  to  violet  is  hardly  appreciable. 


The  same  results  may  be  obtained  by  placing  a  screen  55,  figure  46,  with 
two  parallel  slits,  under  the  vertical  telescope,  to  admit  and  limit  the  two 
rays  c  and  c'  in  figure  44.  It  was  found  that  slits  3  or  even  6  mm.  apart  may 
still  show  fringes,  though  they  are  obviously  smaller  as  the  distance  apart  is 
greater.  The  most  interesting  results  were  obtained  by  bringing  the  rays  a 
little  beyond  the  edge  of  the  prism,  so  that  the  spectra  in  the  telescope  55', 
figure  48,  are  separated  at  some  distance.  A  long  collimator  (i  meter)  is 
advantageous.  In  this  way  the  character  of  these  fringes  was  definitely 
established.  They  are  of  the  elliptic  type,  as  suggested  in  the  figure,  char- 
acteristic of  the  displacement  interferometer,  and  the  cases,  figure  45,  a,  b,  c, 
d,  e,  are  thus  merely  the  intercepts  of  ellipses  with  the  distant  centers, 
between  parallels.  Very  coarse  central  fringes  were  obtained  in  the  dark 
gap,  and  the  displacement  at  mirror  M,  between  the  extreme  hair-line  types, 
was  now  only  about  0.08  cm.,  all  fringes  filing  by  the  coincident  D  lines 
while  the  micrometer  shifted. 

Hence  this  method  is  available  for  displacement  interferometry,  the  hori- 
zontal type  c,  figure  45,  normal  to  the  D  lines,  being  used  for  setting  the  mi- 
crometer at  M.  If  a  plate  of  glass  of  thickness  E  and  index  of  refraction  p. 
is  inserted  normally  into  one  beam,  the  corresponding  air-path  is 


when  B  is  Cauchy's  constant  and  the  wave-length  X.  I  assumed^  =  4.6  X  IQ-U 
and  B/\*  =  0.0265.  On  the  other  hand,  the  same  air-path  for  a  normal  dis- 
placement, e,  of  the  mirror  as  given  by  the  micrometer-screw  is 

X  =  26  COS  (90°  -  0)  /2  =  26  COS  5/2 

where  0  is  the  angle  of  diffraction  of  the  grating  G,  figure  43,  and  b  b'  is  nor- 
mal to  the  direction  of  incident  light.  Hence 

£(M—  I+2J3/X2)  =  26  cos  (90°—  6)/ 2 
A  rough  experiment  was  made  with  a  plate  £  =  2.2  cm.  thick,  ^=1-53. 


REVERSED   AND   NON-REVERSED   SPECTRA. 


67 


The  corresponding  displacement  found  was  0  =  0.92  cm.  The  computed  dis- 
placement should  be  e  =  o.8o  cm.  The  low  value  was  supposed  to  be  due  to 
the  need  of  readjustment  (wedge-shape)  and  insufficient  normality  of  plate. 
Further  data  (table  13)  were  therefore  investigated  for  thinner  plates,  but 
they  do  not  clear  away  the  difficulties. 

TABLE    13. — Inverted  spectra.     0=9°  39'.     x=2e  cos  (90°  —  0)/2. 
^  =  1-526;  5  =  4.6X10-";  25/X2  =  0.0265 ;  z  =  E  (M-i)+2EB/X*. 


E 

Ob- 
served. 

e 

Ob- 
served. 

X 

e 

Computed 
from*  =  2 

z 

E 

Computed 
from  e 
(observed) 
and  x  =  z 

0.736 

0.298 

0.282 
0.295 

0.455 
0.431 

0.451 

0.268 

0.268 
0.268 

0.3901 
+0.0195 
=  0.4096 

0.817 

0.774 
0.809 

o  ^14 

0.480 

0.268 

0.861 

0.311 

0-475 

0.268 



0-853 

In  table  13,  e  is  the  observed  normal  displacement  of  the  mirror  M,  x  the 
corresponding  computed  path-difference,  z  the  path-difference  computed  for 
the  glass  of  thickness  E  and  the  index  of  refraction  p.  From  z  the  displace- 
ment e  or  from  x  the  glass  thickness  E  may  be  computed.  These  are  given 
in  the  table.  In  the  second  and  third  parts  of  the  table,  the  edge  of  the  prism 
P'  was  inclined  at  an  angle  to  the  line  of  symmetry,  in  opposite  directions. 
The  effect  of  this  is  manifest,  but  it  does  not  explain  the  very  large  values  of 
x  for  the  symmetrical  adjustment  (rays  from  grating  to  mirror  making  am 
isosceles  triangle)  of  the  next  observations.  These  were  made  with  care.  P'r 
figure  43,  lay  at  the  middle  of  the  base  of  the  isosceles  triangle  of  rays  from 
G.  The  long  collimator  was  used,  giving  two  splendid  spectra,  and  rays  were 
raised  until  magnificent  large,  cord-like  fringes  appeared.  The  longitudinal 
fringe,  a  single  line  normal  to  the  sodium  lines  extending  throughout  the 
spectrum,  made  it  possible  to  set  the  micrometer  to  about  0.0003  cm.,  or  5 
wave-lengths.  The  motion  of  fringes  with  the  rotation  of  plate  being  very 
striking,  the  plate  was  placed  in  the  normal  position  to  the  rays  by  noting 
the  retrogression  of  fringes  at  this  point.  The  total  range  of  displacement 
between  extreme  hair-like  striations  was  0.25  to  0.30  cm.  One  circumstance 
was  noticed  for  the  first  time — that  on  limiting  the  blade-like  beam  from 
below  (as  above  described)  the  fringes  not  only  enlarge,  but  rotate — i.e., 
path-difference  is  modified. 

It  is  thus  difficult  to  ascertain  the  discrepancies  in  *  in  table  13,  as  these 
values  should  be  nearly  0.41  cm.  throughout,  or  in  e,  which  should  be  about 
0.27  cm.  when  the  prism  is  symmetrically  placed. 

32.  Inverted  spectra.  Further  measurements.— The  curious  results  shown 
in  table  13  induced  me  to  endeavor  to  detect  the  nature  of  the  discrepancy 
by  displacing  the  prism  in  the  direction  of  the  beams  incident  upon  it.  To 


68 


THE   INTERFEROMETRY   OF 


do  this  effectively  it  was  necessary  to  do  the  work  at  long  distances  (meters), 
in  order  that  adequate  space  might  be  available  between  opaque  mirrors 
and  prism.  Accordingly  M,  N,  P',  figure  43,  were  all  placed  on  micrometers, 
with  the  screws  normal  to  the  faces  of  the  mirrors  and  the  right  edge  of  the 
prism,  respectively.  The  fringes  were  found  without  difficulty  and  they 
were  large  and  perfect  near  the  edge  of  contact  of  the  spectra.  Though  the 
sunlight  was  waning,  a  few  measurements  of  ranges  of  displacement  were 
made.  They  were  on  the  average  (e  at  M  and  N,  y  at  P) : 

eM  =  o.og$  cm.  e  AT  =  0.097  cai-  y~  0.062  cm. 

and  since  #=20  cos  5/2  should  correspond  to  2y, 

XM  =  0.145  cm.  ##  =  0.148  cm.  27  =  0. 124  cm. 

Here,  as  above,  x>2y,  or  the  sliding  along  the  edge  of  P  which  accompanies 
e  is  distinctly  effective,  being  nearly  16  per  cent  of  zy. 

Next  day,  with  a  bright  sun,  so  that  much  finer  fringes  could  still  be  de- 
tected, the  range  could  be  increased  to  0^=0.2  cm.  or  #M=°-3  ccn-  when  the 
spectra  were  all  but  separated  on  their  near  edges  and  fringes  very  large. 
For  the  case  of  much  overlapping  of  spectra,  0^=0.15  cm.,  #^=0.22  cm., 
were  obtained.  Finally,  when  the  spectra  were  all  but  separated  on  the  far 
edges  (implying  reflection  at  some  distance  from  the  edge  of  the  prism  P), 
the  fringes  were  glittering,  but  too  small  to  be  distinctly  seen. 

TABLE  14. — Inverted  spectra.    Long  distances.    Plate.   E  =  0.434  cm- ;  A*  =  1-533 ; 
B  =4.6X10-";  z  =  o.23i3+.oi  15  =  0.2428. 


Series,  etc. 

Observed 
e 

Observed 
e 

Observed 

2-y 

X 

X 

I.  Ruled  grating.    io6Z)  —  352 

cm. 
o  184. 

cm. 
o  186 

cm. 

O  25^ 

cm. 

cm. 

o  184 

o  185 

O  2S^ 

Mean 

o  184 

o  185 

o  281 

o  281 

II.  Do.,  another  adjustment.  . 

0.185 
0.185 
0.186 

0.184 
0.184 

0.246 
0.246 

o  24.8 

0.186 

O  24.8 

Mean 

o  185 

o  283 

o  281 

III.  Do.,  another  adjustment.. 

o  24.4. 

IV.  Film  grating.    10*0  =  167. 
V.  Prism  60°  



0.244 
0.234 
0.240 
O  24.4, 

In  table  14  the  results  for  eM ,  eN,  and  y  are  given,  when  these  displacements 
are  produced  by  a  glass  plate  £=0.434  cm.  thick.  If  the  coefficient  of  dis- 
persion B  is  assumed,  the  displacement  computed  from  X,  n,  E  would  be 
2=0.243  cm.  Although  the  values  of  zy  are  larger  than  this,  the  difference 


REVERSED   AND   NON-REVERSED   SPECTRA.  69 

must  be  ascribed  to  the  assumed  value  of  B  and  to  the  difficulty  of  placing 
the  plate  normal  to  the  rays.  The  method  of  reversion  of  fringes,  used  else- 
where, is  not  sufficiently  sensitive  here.  The  data  for  x  are  again  in  excess 
of  y  by  about  13  per  cent. 

Finally,  a  series  of  consecutive  measurements  were  made  of  the  ranges  of 
displacement,  zy,  for  different  dispersive  powers  at  G,  figure  43,  all  under 
(otherwise)  like  circumstances.  The  mean  results  were  for 

60°  prism d6/d\=  760.         2y  =  o.o6ocm. 

Ruled  grating 2,880.  .117 

Film  grating 6,400.  .208 

Three  or  more  measurements,  completed  in  each  case,  were  in  good  agree- 
ment. The  attempt  was  also  made  to  use  a  45°  prism  here,  but  the  spectra 
were  too  small  and  the  fringes  could  not  be  found.  In  each  of  these  cases 
the  value  of  2y  for  the  plate  £=0.434  cm.,  as  shown  in  table  14,  series  III, 
IV,  V,  are  virtually  the  same.  The  rapid  increase  of  range  of  displacement 
with  the  dispersion  of  the  system  is  thus  again  encountered. 

33.  Rotation  of  fringes. — A  word  must  now  be  given  relative  to  the  rotation 
of  fringes,  which  is  here  throughout  180°,  whereas  in  the  similar  case  above 
(Chapter  I,  §§  25,  26)  the  rotation  was  but  90°.  It  will  be  seen  on  consulting 
figure  43  that  if  M  moves  micrometrically,  normal  to  itself,  the  pencil  b  will 
slide  fore  and  aft,  along  the  edge  of  the  reflecting  prism  P'.  Thus  6  may  be 
either  in  front  of  or  behind  b'  or  coplanar  with  it  in  a  vertical  plane.  It  will 
not  generally  be  collinear.  This  is  an  essential  part  of  the  explanation. 


<8      4 


In  figure  49,  let  a  and  b  be  the  two  patches  of  light  of  like  color  and  origin, 
which  produce  interferences.  The  fringes  will  therefore  be  arranged  in  the 
direction  /,  normal  to  the  line  ab.  Now  suppose  a  is  moved  toward  the  right 
or  b  toward  the  left,  or  both,  parallel  to  the  edge  of  the  prism,  as  the  arrows 
in  the  figure  suggest.  Then  the  fringes  will  successively  take  the  trends  of 
which  cases  i,  2,  3,  4  are  typical  examples.  In  other  words,  they  will  be 
markedly  accelerated  and  retarded  in  passing  through  the  cases  2  and  3  re- 
spectively. This  is  precisely  what  takes  place  and  suggests  why  the  case 
between  2  and  3  may  be  used  as  a  fiducial  mark  in  interferometry.  If  a  and 
6  also  move  vertically,  in  figure  49  there  will  be  no  essential  difference,  unless 
the  latter  motion  is  large.  In  such  a  case  the  rotation  may  become  1,2,2,1. 

The  displacement  of  6  parallel  to  itself,  for  a  normal  displacement  e  of  the 
mirror  M,  will  be,  as  above,  figure  17,  if  5=0'—  6  =  90°— 9 
s  =  2e  sin  <J/2 


70  THE   INTERFEROMETRY  OF 

and  the  corresponding  displacement  of  c  parallel  to  itself,  since  ^'  =  90°, 
t  =  s  tan  (f>'/2=s  =  2e  sin  d/2. 

If  0=9°39',  t=s  =  2eXo.8i  =  1.626.  Thus  if  0  =  0.25  cm.,  2  =  0.4  cm.,  and 
since  /  is  twice  the  width  of  the  patches  or  strips  sliding  over  each  other,  the 
width  of  this  strip  would  be  0.2  cm. 

It  is  the  sliding  of  the  pencil  b  along  the  edge  of  P'  which  introduces  addi- 
tional path-differences  whenever  P1  is  not  symmetrical  and  its  edge  not  par- 
allel to  the  plane  a  a'.  It  is  probable  also  that  the  same  restrictions  as  to 
the  breadth  and  depth  of  efficient  wave-fronts  will  apply  here  as  before;  but 
this  should  be  specially  investigated.  The  nearly  circular  outline  of  the  locus 
of  fringes,  when  spectra  are  both  reversed  and  inverted  in  §36,  even  though 
homogeneous  light  is  in  question  in  the  last  case,  clearly  points  in  this  direction. 

34.  Range  of  displacement  varying  with  orientation  of  reflector  P'. — The 

displacement  e  of  the  mirrors  M  and  N  slides  the  corresponding  pencil  b 
or  br,  figure  43,  along  the  edge  of  the  reflecting  prism  P',  and  a  reason  for 
the  rotation  of  fringes  is  thus  easily  at  hand.  It  does  not  at  once  appear  why 
the  right  or  the  left  displacement  (y)  of  P'  should  also  produce  a  rotation  of 
fringes ;  for  here  the  pencils  b  and  b'  remain  collinear  and  there  is  no  sliding. 
It  must  thus  be  remembered,  however,  that  the  fringes  are  ultimately  ellip- 
tic; for  the  axis  parallel  to  the  Fraunhofer  lines  is  conditioned  by  the  obliquity 
of  rays  in  this  plane  only,  whereas  the  axis  in  the  direction  of  the  length  of 
spectrum  depends  on  dispersion.  The  motion  y  of  P'  displaces  these  ellipses 
bodily  through  the  spectrum.  Hence  the  fringes  first  appear  at  any  given 
Fraunhofer  line  in  the  form  of  hair-like  striations  parallel  to  it.  These 
enlarge  and  rotate  to  a  maximum  normal  to  the  Fraunhofer  line.  In  fact,  a 
single  interference  line  may  now  run  from  end  to  end  of  the  spectrum.  There- 
after the  fringes  vanish  in  symmetrically  the  same  manner  and  are  last  seen 
as  fine  striations  parallel  to  the  Fraunhofer  line.  It  is  possible,  therefore, 
that  the  sliding  of  pencils  which  accompanies  the  e  displacement  accounts 
for  the  difference  of  values  of  x  =  ze  cos  d/2  and  2y. 

Before  discussing  this  question  further  it  seemed  necessary  to  study  the 
effect  of  different  orientations  of  P'  relative  to  the  b  b'  rays.  One  may  note, 
preliminarily,  that  a  rotation  of  M  and  AT  on  a  horizontal  axis  parallel  to 
their  faces,  or  of  P'  on  a  horizontal  axis  parallel  to  its  edge,  also  rotates  the 
fringes;  but  it  seems  probable  that  these  motions  are  virtually  equivalent  to 
a  displacement,  y,  of  the  edge  of  the  prism. 

The  fringes  may  be  seen  in  all  focal  planes,  at  least  the  long  line  parallel 
to  the  length  of  spectrum.  The  others  may  often  be  restored  by  rotating  the 
grating.  The  marked  occurrence  of  fringes  in  the  narrow  longitudinal  gap 
between  two  spectra  (overlapping  just  removed  by  the  rotation  of  M  and  N 
on  a  horizontal  axis)  can  possibly  be  explained  in  this  way.  These  fringes  in 
the  dark  space  are  very  sharp  and  luminous  and  seen  in  the  principal  focal 
plane  with  the  Fraunhofer  lines.  But  it  will  usually  be  found  that  on  draw- 
ing the  ocular  out  the  separated  spectra  will  overlap  at  their  edges  again, 


REVERSED  AND   NON-REVERSED   SPECTRA.  71 

whereas  on  pushing  the  ocular  in  from  the  principal  position  the  separation 
is  increased.  Hence,  to  account  for  the  disturbance  in  the  ether  gap,  as  it 
were,  it  seems  most  reasonable  to  assume  that  the  rays  cross  and  interference 
occurs  after  the  rays  have  passed  the  principal  focal  plane  (i.e.,  nearer  the 
eye  of  the  observer),  and  that  the  interferences  occurring  here  are  projected 
into  the  principal  focal  plane.  Nevertheless,  the  fringes  are  so  strong  and 
sharp  that  the  two  clearly  focused  spectra  seem  to  react  on  each  other  across 
the  gap  at  their  edges.  I  have  pointed  out  similar  phenomena  in  the  pre- 
ceding report  (Carnegie  Inst.  Wash.  Pub.  249).  The  case  is  just  as  if  a  tele- 
scope or  lens  focused  on  a  single  or  a  Young's  double  slit  (with  the  images 
sharply  delineated)  should  show  fringes. 

In  adjusting  the  interferometer,  figure  43,  for  these  experiments,  the  fol- 
lowing systematic  plan  was  pursued.  By  a  rough  adjustment  with  sunlight 
and  measurement,  all  parts  of  the  apparatus  are  first  placed  symmetrically 
to  each  other  (as  in  figure).  The  direct  beam  should  just  graze  the  edge  of 
the  prism  P'  and  the  naked  eye,  viewing  the  edge  from  above,  should  see  the 
two  bright  rays  of  the  same  color  (reflected  from  M  and  N)  contiguously 
near  the  edge.  In  the  spectra  of  the  same  length  on  the  two  sides  of  the 
prism  P',  the  same  colors  are  opposite  each  other.  On  looking  down  on  the 
edge  of  the  prism  with  a  telescope  (fine  slit),  two  sharp  and  clear  spectra 
should  be  seen,  which  can  be  made  to  overlap  at  their  edges  in  any  amount 
by  rotating  M  and  N  on  a  horizontal  axis.  Finally,  the  D  lines  are  brought 
into  coincidence  by  rotating  the  grating  G  on  a  vertical  axis.  The  largest 
fringes  are  obtained  by  slightly  raising  and  lowering  the  incident  beam  L  to 
the  grazing  position  in  question.  By  displacing  P  on  the  micrometer,  right 
and  left,  the  fringes  are  soon  found. 

The  first  experiments  were  made  with  the  object  of  testing  the  effect  of  a 
slant,  to  the  right  or  left,  of  the  edge  of  the  prism  on  the  range  of  displace- 
ment y.  With  a  60°  prism  at  P,  the  search  was  found  to  be  too  difficult  and 
therefore  soon  was  given  up.  The  few  data  obtained  were :  Edge  of  P' sym- 
metrical— range,  y=o.o6 1,  0.050,  0.062  cm.;  edge  of  P'  toward  left — range, 
^=0.63  cm.;  showing  no  certain  difference. 

P  was  therefore  replaced  by  a  ruled  grating  (£  =  352X10-*  cm.)  to  obtain 
greater  dispersion.  The  ranges  of  displacement,  y,  now  found  were:  edge  of 
P'  to  left,  y=o.i2o,  0.130  cm.;  edge  of  P'  symmetrical,  ^=0.128,  0.127  cm.; 
edge  of  P'  to  right — y  =  o.no,  0.113  cm. ;  readjusted,  0.130,  0.132  cm.  These 
differences  are  apparently  incidental,  as  much  depends  on  the  light.  In  a 
darker  field  fringes  vanish  sooner.  One  may  assume  that  slight  inclinations 
of  the  edge  of  the  reflecting  prism  P'  are  without  consequence. 

The  next  experiment  was  to  determine  the  effect  of  a  lack  of  collinearity 
of  the  rays  b  b'.  This  shows  itself  to  the  naked  eye  looking  down  upon  the 
edge  of  the  prism  from  above,  since  by  rotating  M  and  N  in  contrary  direc- 
tions around  a  vertical  axis  the  bright  spots  of  light  move  along  the  edge  of 
the  prism  from  front  to  rear,  or  the  reverse.  If  regarded  by  a  telescope, 
the  axis  of  the  instrument  will  be  correspondingly  inclined  toward  the  front 


72  THE   INTERFEROMETRY  OF 

or  to  the  rear.  The  data  obtained  for  the  range  of  displacement  were  now: 
Images  and  telescope  toward  rear,  ^  =  0.124,  0.140  cm.;  ^=0.150,  0.142  cm.; 
images  and  telescope  toward  front,  y= 0.146  cm.,  ?  =  0.154  cm. 

These  differences  are  again  incidental.  The  following  data  were  subse- 
quently found:  Telescope  inclined  rearward,  y=o.ioo,  o.no  cm.;  0.143, 
0.140  cm.;  telescope  vertical,  ^=0.150,  0.150  cm.;  0.162,  0.151  cm.;  tele- 
scope inclined  forward,  ^=0.150,  0.133  cm.;  0.145,  0.162  cm. 

In  the  first  experiment  the  illumination  was  insufficient,  so  that  the  finer 
fringes  escaped  detection.  Hence,  it  is  here  also  probable  that  slight  depar- 
ture from  collinearity  in  the  rays  b  b',  normal  to  the  edge  of  the  prism  P't  is 
without  consequence.  Discrepancies  are  introduced  by  changes  in  the  in- 
tensity of  light — conditions  which  are  often  hard  to  control. 

As  the  range  of  displacement  is  not  a  quantity  which  can  be  accurately 
ascertained,  the  effect  of  the  insertion  of  a  glass-plate  compensator,  0.434  cm. 
thick,  was  determined,  with  a  similar  end  in  view,  for  different  angles  of  the 
rays  b  b'  (nearly  normal)  to  the  edge  of  the  prism  P'.  The  results  were : 
Telescope  inclined  toward  front,  y  =  0.122,  0.123  cm.;  telescope  vertical, 
y  =  0.122,  0.122,  0.122  cm.;  telescope  inclined  toward  the  rear,  ;y  =  0.120, 
0.12 1  cm.  These  are  the  differences  of  the  displacement  corresponding  to 
the  linear  central  fringes  normal  to  the  sodium  lines,  obtained  in  the  presence 
and  absence  of  the  plate.  The  path-difference  computed  above  was  z—  0.2428 
cm.  This  is  as  near  zy  as  the  observations  warrant. 

It  follows,  therefore,  even  if  the  observations  are  in  their  nature  not  very 
precise,  that  if  the  rays  b  and  b'  meet  at  such  small  angles  as  any  reasonable 
adjustment  may  introduce,  the  effect  may  be  disregarded.  Furthermore, 
that  the  difference  between  x  =  2e  cos  5/2  (where  e  is  obtained  by  moving 
the  mirrors  M  and  N  parallel  to  themselves)  and  zy  (obtained  by  moving 
P'  at  right  angles  to  its  edge)  is  to  be  ascribed  to  the  sliding  along  or  across  the 
edge.  The  rotation  of  fringes  which  necessarily  occurs  in  displacement  inter- 
ferometry  by  the  shifting  of  the  ellipses  is  augmented  or  decreased  in  the 
former  case  (x)  by  the  equivalent  of  the  sliding  in  question.  The  reason  for 
this  has  clearly  been  suggested  in  connection  with  figure  18,  Chapter  I, 
and  figure  49,  Chapter  II. 

35.  Range  of  displacement  varying  with  dispersion.— The  interesting 
method  in  Chapter  I,  §  25,  where  the  opaque  mirrors  are  replaced  by  two 
identical  gratings  (halves  of  the  same  grating)  with  the  object  of  obtaining 
successive  orders  of  dispersion,  may  be  used  in  connection  with  figure  43  of 
the  present  chapter.  It  is  therefore  the  object  to  find  the  range  of  displace- 
ment y  of  the  prism  P'  when  the  fringes  pass  from  the  initial  transverse 
hair-lines  to  the  final  transverse  hair-lines  (fig.  45),  through  the  longitudinal 
maximum  of  size.  The  same  difficulty  inheres  in  this  method  as  in  the  above, 
viz,  it  is  not  possible  to  state  precisely  when  the  hair-lines  have  vanished; 
but  the  successive  orders  of  range  of  displacement  are  so  different  that  inter- 
pretable  results  are  obtained.  The  experiments  in  the  large  interferometer 


REVERSED  AND   NON-REVERSED   SPECTRA.  73 

proved  very  trying,  however,  because  the  ruled  faces  of  the  available  grat- 
ings at  M  and  N  were  but  0.5  inch  square.  There  is  thus,  without  refined 
and  special  instrumental  equipment,  considerable  difficulty  in  adjusting  the 
rays  to  this  small  surface.  This  was  particularly  true  in  the  higher  orders  of 
spectra. 

To  obtain  sufficient  light  the  resolving  grating  G  was  replaced  by  a  60° 
prism.  The  dispersive  powers  are  thus  the  same  as  in  §  25,  Chapter  I.  The 
work  proceeded  smoothly  in  the  orders  of  o  (reflection  from  grating  face) 
and  i.  In  the  third,  the  fringes  were  hard  to  find  and  hard  to  retain,  for  rea- 
sons which  I  do  not  understand.  There  was  abundance  of  light,  except  in 
the  fourth  order,  which  was  abandoned  for  that  reason.  The  best  results 

Order  o,  y  =  0.066  cm.        d6/d\  =    760 
if          -23<>  3,500 

%  2,  .450  6,400 

3 1  -650  9,9OO 

Though  much  time  was  spent  on  this  work,  the  results  (excepting  the 
first)  are  doubtless  still  too  low.  Since  the  path-difference  zy  corresponds 
to  x  =  2e  cos  8/2,  cceteris  paribus,  and  since  the  data  in  x,  table  12,  are  essen- 
tially half  the  total  range  (rotation  but  90°,  while  it  is  180°  here),  y  corre- 
sponds to  x.  Thus  the  results  in  table  12  are  larger  throughout,  but  the 
present  data  make  a  smoother  series  even  through  the  third  order. 

36.  Spectra  both  reversed  and  inverted. — This  is  an  interesting  combina- 
tion of  the  two  methods  of  investigation  and  not  very  difficult  to  produce. 
Retaining  the  adjustment  for  inverted  spectra  as  in  §30,  figure  43,  the  light 
impinging  on  the  grating  G  is  dispersed,  preferably  by  a  direct-vision  grating 
(with  auxiliary  prism).  The  rulings  of  both  gratings  (the  prism  grating  g 
inserted  as  shown  being  between  the  collimator  at  some  distance  and  the 
grating  of  the  interferometer  G,  figure  43)  must  be  parallel.  If  the  grating 
constants  are  different  (D  =  167X10-*  cm.  film  and  .0  =  352X1  cr6  ruled 
grating  were  employed),  the  spectra  in  the  telescope  are  naturally  of  different 
lengths;  for  the  dispersion  of  the  prism  grating  g  is  increased  on  one  side  and 
decreased  on  the  other  side  by  the  second  grating  G.  Moreover,  this  decrease 
from  the  larger  dispersion  of  the  first  grating  g  is  beyond  zero  (achromatism) 
into  negative  values.  Hence,  the  corresponding  duplicate  spectrum  in  the 
telescope  is  a  small  and  a  large  spectrum  reversed,  while  the  inversion  re- 
mains intact.  In  the  experiment  made,  the  larger  D\D'z  distance  was  some- 
what more  than  twice  the  smaller  AA- 

It  is  now  merely  necessary  to  place  any  longitudinal  axis  (line  of  symmetry) 
of  the  spectra  in  contact,  or  it  is  but  necessary  that  the  spectra  are  longitud- 
inally parallel  and  overlap.  The  phenomenon  a,  figure  50,  then  appears 
at  the  intersection  of  the  lines  of  longitudinal  and  of  transverse  symmetry. 
It  is  thus  proportionately  nearer  the  smaller  A  A  and  farther  from  the 
larger  D\D'Z  doublets,  but  always  between  them.  If  the  AA  He  within 
the  D'iD'z  lines,  the  fringes  lie  within  the  AA  Pair- 


74 


THE   INTERFEROMETRY  OF 


The  phenomenon,  which  should  be  observed  with  a  powerful  telescope, 
usually  consists  of  three  small  elongated  dots,  lying  within  an  elliptic  locus, 
the  locus  usually  having  a  transverse  axis  (parallel  to  the  Fraunhofer  lines) 
about  two  or  three  times  as  long  as  the  longitudinal  axis  (parallel  to  lengths 
of  spectra).  As  a  rule,  the  width  was  DiD2  and  the  length  larger  than  D\D'Z, 
but  this  ratio  may  be  changed,  as  above,  by  screening  off  the  wave-front. 
The  fringes  are  not  more  than  one-half  of  DiD2  apart  and  are  frequently 
horizontal  (longitudinal),  however  the  micrometer  at  M  is  shifted. 

The  interesting  result  is  here  again  met,  incidentally,  that  spectra,  though 
of  different  lengths,  are  nevertheless  quite  capable  of  producing  strong  inter- 
ferences. 

a 


50 


51 


In  further  experiments  with  the  long  collimator  and  very  bright  spectra,  a 
variety  of  other  forms  were  obtained,  shown  at  b,  c,  d,  figure  50.  In  the  patterns 
a  and  6,  the  elliptic  outline,  sometimes  circular,  is  always  evident  from  the 
enhanced  brightness  of  the  bright  fringes  of  the  spot.  The  arrow-shaped 
form,  c,  inclosed  a  bright  egg,  whereas  d  was  usually  sharply  semicircular. 
As  any  adjustment  of  overlapping  spectra  suffices,  the  D  lines  may  be  quite 
out  of  the  field,  or  the  spectra  may  be  slightly  separated  with  the  interfer- 
ence spot  in  the  gap. 

The  experiment  was  also  made  of  crossing  the  spectra  at  some  other  angle 
than  o°  or  180°.  To  do  this  the  rulings  of  the  prism  grating  were  placed  at 
right  angles  to  those  of  the  interferometer  grating,  as  in  Newton's  method  of 
crossed  prisms.  Seen  in  the  telescope  (adjusted  for  inverted  spectra,  as 
above,  §  30)  the  two  spectra  now  made  an  elbow  with  each  other,  figure  51, 
while  the  AA  lines  are  still  parallel  and  can  be  put  in  coincidence.  At 
first  no  interferences  could  be  detected  in  any  adjustment.  Later,  however, 
on  using  the  large  collimator,  strong  interferences  were  obtained  in  the  line 
of  symmetry  of  the  elbow  and  normal  to  the  D  lines,  as  shown.  They  have 
the  same  characteristics  as  the  preceding  and  persist  during  a  displacement 
of  M  of  about  0.3  cm. 

37.  Experiments  with  the  concave  grating.— If  in  the  device  figure  40, 
Chapter  I,  the  prism  P  is  rotated  90°  on  an  axis  parallel  to  b  b',  so  that  the 
rays  move  upward,  the  phenomenon  of  inverted  spectra  may  be  realized. 
The  fringes  are  observed  with  a  lens  from  above  or  reflected  forward.  They 


REVERSED  AND   NON-REVERSED   SPECTRA.  75 

were  found  without  much  difficulty  and  showed  a  range  of  displacement  (y) 
of  the  prism  P,  right  or  left,  in  various  adjustments,  of  at  least  ^=0.71,  0.61, 
0.67  cm.  larger,  therefore,  than  with  the  forward  prism,  as  was  inferred.  Of 
course,  much  depends  upon  how  far  the  extremely  fine  fringes  at  the  begin- 
ning and  end  are  pursued. 

This  method  is  not  very  convenient  for  the  present  purposes.  In  the  first 
place,  the  distance  GMPF  is  given,  an  unnecessary  restriction  on  the  adjust- 
ment, unless  the  lens  T  is  replaced  by  a  short-distance  telescope,  which  has 
other  disadvantages.  In  the  second  place,  the  mirrors  M  and  N  can  not  be 
used  for  displacement,  as  they  move  the  Fraunhofer  lines  of  the  correspond- 
ing spectrum.  In  fact,  M  or  N  affords  a  method  for  the  adjustment  of 
these  lines  to  coincidence.  Finally,  the  amount  of  overlapping  which  is 
usually  secured  is  always  very  partial,  and  if  the  edge  of  the  prism  P  is  not 
quite  sharp  and  well  silvered,  the  edges  are  ragged.  Even  in  the  right-and- 
left  displacement  (y)  of  P,  the  spectra  are  carried  bodily  with  it  in  front  of 
T.  Although  this  is  no  serious  objection,  it  is  an  unnecessary  complication. 
In  spite  of  the  brilliant  spectra  and  large  ranges,  I  did  not  spend  much  time 
in  developing  the  method. 

38.  Conclusion.  General  methods. — The  origin  of  the  phenomenon  of 
reversed  spectra  seems  to  be  the  slit  of  the  collimator,  the  diffraction  of  which 
furnishes  a  patch  of  light,  effectively  i  or  2  mm.  in  breadth,  out  of  which  the 
component  rays  are  to  be  separated.  Spectroscopically  the  slit  furnishes  the 
degree  of  homogeneous  light  within  which  the  phenomenon  may  be  developed. 

In  the  case  of  inverted  spectra,  the  slit  is  not  primarily  necessary,  for  here 
the  interferences  occur  in  the  direction  of  (or  the  fringes  lie  normally  to) 
the  Fraunhofer  lines  and  therefore  virtually  in  homogeneous  light.  The  fringes 
are  due  to  the  continuous  changes  of  the  obliquity  of  rays  in  each  separate 
color  and  thus  belong  to  the  phenomenon  of  a  wide  slit  with  homogeneous 
light.  The  fringes  of  reversed  spectra  owe  their  occurrence  to  the  continuous 
change  of  the  obliquity  of  rays  produced  by  dispersion  and  require  a  spectro- 
scopic  slit.  In  the  case  of  combined  inversion  and  reversion, the  locus  of  fringes 
is  not  far  from  circular,  though  the  major  axis  of  the  ellipse  corresponds  to 
the  inversion.  Obliquity  and  dispersion  are  thus  about  equally  effective. 

The  patch  of  light  rays  of  identical  origin  may  now  be  separated  into  two 
component  rays  by  a  variety  of  methods.  The  white  pencil  may  simply  be 
cleaved  by  the  edge  of  a  sharp  silvered  prism,  or  the  pencil  may  be  refracted 
into  two  beams  at  a  blunt-edged  prism,  or  the  separation  may  be  produced 
by  the  diffraction  of  a  grating,  by  polarization,  etc. 

Two  entirely  distinct  pencils  are  thus  obtained,  subject  to  independent 
control,  by  which  the  phenomenon  of  diffraction  may  be  generalized.  In 
other  words,  in  the  classical  experiments  in  diffraction,  the  diffracting  system 
is  rigid.  Take,  for  instance,  the  following  experiment,  which  has  a  close 
bearing  on  the  phenomenon  of  this  paper.  In  figure  52,  L  is  a  distant  slit  or 
a  fine  Nernst  filament,  P  the  principal  plane  of  the  objective  of  a  telescope 


76  THE   INTERFEROMETRY  OF 

with  the  principal  focus  at  F,  to  be  observed  with  the  eye-piece.  The  screen 
5  5'  with  a  double  slit  parallel  to  the  linear  source  L  is  placed  in  front  of  the 
objective.  Brilliant  interferences  are  then  seen  at  F,  which  are  coarser  as 
the  distance  e  between  the  slits  5  s'  is  smaller,  and  if  the  distance  PF  is  r  and 
the  distance  between  fringes  is  x,  the  usual  equation 

\/e=x/r 

is  applicable.  Owing  to  the  direct  application  of  this  experiment  to  the  above 
investigations,  its  very  fundamental  importance  in  the  theory  of  resolution 
of  optical  instruments,  etc.,  it  seemed  worth  while  to  give  it  experimental 
treatment  here.  The  screen  s  s'  is  conveniently  made  by  cutting  parallel 
lines  with  a  sharp  triangular  cutting  stylus  and  a  steel  T-square,  on  a  black- 
ened gelatine  dry  plate.  A  number  of  such  doublets  0.05  to  0.5  cm.  apart 
may  be  ruled  at  a  distance  of  about  0.5  inch  apart.  When  the  parts  are  as- 
sembled, the  fringes  may  be  seen  in  all  focal  planes  F,  and  the  fringes  are  in 
fact  much  more  brilliant  with  the  ocular  out  of  focus.  The  enormously  large 
diffraction  of  each  single  slit  is  simultaneously  visible,  and  if  two  doublets 
are  close  enough  together,  their  systems  may  be  seen  superposed.  If  c  is  small 
(less  than  o.i  cm.)  the  fringes  are  in  fact  visible  to  the  naked  eye  without  a 
telescope.  With  two  identical  doublets  close  together,  the  fringes  may  be 
seen  to  be  alternately  in  step  and  out,  as  the  ocular  of  the  telescope  moves 
outward,  until  finally  the  diffractions  of  the  "rod"  between  the  doublets  is 
strikingly  manifest.  This  has  also  been  generalized  in  the  present  work. 


In  all  these  classical  cases  there  is  a  continuous  succession  of  pairs  of  corre- 
sponding points  (one  of  the  pair  in  each  slit  of  the  doublet)  between  which 
interferences  occur.  The  line  of  any  two  such  points  is  rigidly  normal  to  the 
direction  of  the  slits.  In  the  above  experiments  with  spectra,  however,  the 
two  points  may  not  only  have  any  relation  to  each  other,  but  either  point 
may  be  moved  at  pleasure.  This  gives  rise  to  the  bewildering  variety  of 
beautiful  phenomena,  some  of  them  useful,  which  I  have  tried  to  describe 
in  the  present  and  preceding  reports. 

With  the  beams  of  like  origin  separated,  it  is  next  necessary  to  bring  them 
together  again.  This  requires  at  least  one  independently  controllable  reflec- 
tion for  each  beam.  In  the  interesting  group  of  phenomena  obtained  with 
crossed  rays,  two  reflections  may  be  desirable,  though  with  a  change  of  appa- 
ratus a  single  reflection  here  also  suffices.  Thereafter  the  beams  may  be  com- 
pounded in  a  manner  inverse  to  the  one  by  which  they  were  produced,  for 
instance,  by  the  reflection  of  a  silvered  obtuse  prism,  by  refraction  toward 
the  edge  of  a  prism,  by  a  grating,  by  polarization,  etc.  To  observe  the  recom- 


REVERSED  AND   NON-REVERSED   SPECTRA. 


77 


bined  ray,  the  telescope  is  always  the  more  convenient  instrument,  since  its 
use  is  not  restricted  to  definite  distances  from  the  system. 

It  has  been  shown  that  as  a  whole  the  above  phenomena  correspond  very 
closely  to  the  behavior  of  the  ellipses  encountered  in  displacement  interferom- 
etry.  Thus  the  cases  of  non-reversed  spectra,  of  inverted  spectra,  etc.,  if  we  dis- 
regard certain  exceptional  accompaniments  for  the  moment,  can  be  at  once  so 
classified.  The  shift  of  ellipses  behind  a  narrow  slit  in  an  opaque  screen  and  ob- 
served in  front  of  it,  for  instance,  would  exhibit  all  the  rotational  occurrences. 

39.  Displacement  interferometry.  Equations.  —  It  is  thus  desirable  to  ad- 
duce the  equations  of  displacement  interferometry  in  a  somewhat  different 
way,  but  in  the  main  as  taken  from  my  earlier  reports.  In  figure  53,  G  is  a 
thick  plate  of  glass  on  which  the  blade- 
shaped'  pencil  of  white  light  L  from  a 
collimator  impinges  at  an  angle  i.  M 
and  N  are  the  opaque  mirrors  of  a 
Michelson  device.  G'  is  a  plate  grating 
by  which  the  white  beam  R  (feeble 
spectrum)  is  resolved,  P  the  principal 
plane  of  the  objective  of  the  telescope, 
ro  the  image  seen  through  the  ocular. 
The  plate  disperses  the  white  light  L 
into  the  spectrum  rv,  as  shown  in  the 
figure.  The  direct  reflection  at  R'  is 
not  used.  For  convenience  in  discus- 
sion the  mirror  N  and  its  component 
ray  may  be  rotated  180°  around  the 
trace  of  the  grating  G  as  an  axis,  into 
the  position  N',  where  IN  becomes  x+ 

>'t  intercepted  between  normals.    If  perpendiculars  be  let  fall  from 


7  to  AT  or  N'  and  /'  to  M,  their  difference  of  length  is 

(1)  N=x+x"  =  ecosi+x" 

N  is  an  important  coordinate  used  throughout,  below,  since  it  is  indepen- 
dent of  color  (X  and  M).  In  (i)  *  is  the  angle  of  incidence,  R  of  refraction 
of  the  plate  of  thickness  e  and  index  of  refraction  M- 

If  we  draw  the  wave-front  w,  the  path-length  of  the  red  ray  through  glass  to 
M,  for  instance,  is  en/cos  R+p.  The  path  of  the  ray  through  air  (only)  to 
mirror  M  is 

x+x'+p+x"=e  cos  i+e  sin  i  tan  R+p+x" 

If  sin  i=/i  sin  1?  be  introduced,  the  path-difference  thus  becomes,  after  re- 
duction, 

(2)  n\  =  2N-2encosR 
or 

(3)  n\  =  2e  (ju-cos  (i-R))  /cos  R 


78  THE   INTERFEROMETRY   OF 

the  first  equation  being  more  practical.  N,  therefore,  is  the  differences  in 
distances  of  the  extremities  I,  I'  of  the  normal  n  at  the  point  of  impact  I 
from  the  mirrors  N  and  M  respectively;  x"  is  the  air-distance  apart  of  the 
two  mirrors  (after  rotation). 

To  find  the  change  of  wave-length  per  fringe,  dK/dn  may  be  deduced. 

rfX  =  _  X2  _ 
2  dn      N-e(ii  cos  R+\(d[i/dX)  /cosfl) 
This  equation  has  a  maximum  when 


and  this  is  the  coordinate  Ne  for  centers  of  ellipses  on  any  given  spectrum 
line  X.    N=Ne  in  equation  (2)  gives 


da     —  zB 
In  general  (J.=A+B/\2;  —  =  is  adequate  for  experimental  work.  Thus 

Ct\  A 

if  5  =  4.6X10"",  e=i  cm.,  R  =  o,  then  «c  =  88o,  or  880  sodium  wave-lengths 
are  expended  in  the  path-difference  at  the  elliptic  center. 

If  n'  fringes  pass  at  a  given  color  X  for  a  displacement  AN  of  the  mirror  M 

(7)  X  =  2AAT/«' 

If  n'  fringes  lie  between  two  given  colors  X  and  X'  for  the  same  position  N 
of  the  micrometer  mirror  M 


or  if  5  is  the  differential  symbol 

ucosR  i 

w'  =  205-  --  2^5  - 

X  X 

The  question  next  of  interest  is  the  change  of  the  angle  of  altitude  of  the 
ray  per  fringes  transversely  to  the  spectrum.  Light  is  homogeneous,  N  there- 
fore constant.  Let  a  be  the  angle  of  altitude  of  an  oblique  ray  in  the  hori- 
zontal plane  L,  figure  53,  impinging  at  I.  If  i'  and  Rr  are  the  corresponding 
angles  of  incidence  and  refraction,  then  if,  i,  a  make  up  the  sides  of  a  spherical 
triangle,  right-angled  at  the  angle  opposite  i'.  Hence 


(9)  cos  i' = cos  i  cos  a,  and  (i  cos  Rr  =  \/ff  —  i  +cos2  *  cos*  a 

With  this  introduction  of  Rr  for  R,  equation  (2)  is  again  applicable,  since 
nothing  has  been  changed  in  N,  e,  ft,  X,  or  i.    Hence 


(10)  n\  =  zN— 20vV—  i  +  cos2  *  cos2  a 


REVERSED  AND   NON-REVERSED  SPECTRA.  79 

To  determine  the  change  of  altitude  per  fringe, 


da _XVM*~  i -j-  cos1  i  cos1  a 
dn  e  cos2  i  sin  2a 

and  this  is  a  maximum  when  a=o;  i.e.,  in  the  plane  of  figure  53.  In  this 
case  equation  (2)  is  reproduced  from  (10),  so  that  a  double  maximum  occurs 
for  a  =  o  and 


The  other  practical  datum  is  the  shift  of  a  given  fringe  per  centimeter  of 
displacement  of  the  mirrors.  Here  n  and  e  are  constant  while  N,  X,  ^,  R  vary, 
so  that 

(12)     '  —  -  2 

dN  ze    df.i 


2e     du 

This  is  a  maximum  when  n  =  nc=  ---  =4£5/X3,  or  when  N=Ne.    In 

cos  R  d\ 

other  words,  there  is  a  minimum  displacement  relative  to  wave-length  shift 
of  fringe  at  the  centers  of  ellipses. 

Equation  (10)  is  thus  inclusive.    If  i  =  o,  which  is  nearly  the  case,  experi- 
mentally, in  my  work  and  no  restriction  on  the  apparatus,  and  since  a  is 
always   very   small,   equations    (10),    (n),    (12),   etc.,   may  be  simplified. 
Hence  approximately  (i  =  o) 
(13) 


(14)  —  =  -  \V-a2 

dn      tea 

(  <&_  X2 

2dn 


(6^  — 

dN 

where  n  is  the  order  of  the  fringe  at  X  for  N,  e,  /f/,  a.    Again,  ne=  -2e.d[t/d\. 
Equation  (13)  admits  of  an  interpretation  in  terms  of  the  approximately 
elliptic  locus  found  for  constant  n.    The  equation  may  be  written,  if  fi  is 
treated  as  a  mean  constant, 


Here  e  (a/fi)  and  X  may  be  regarded  as  the  coordinates  of  a  curve  described  on 
the  faceof  theplateof  glass  toward  the  observer,  so  that  the  equation  is  an  ellipse 
referred  to  an  eccentric  axis  of  ordinates.  The  axes  of  this  ellipse  are  tqp/N 
horizontally  and  e  vertically.  Of  course,  the  telescope  converges  all  this  to  a 


80  THE   INTERFEROMETRY  OF 

single  white  image  of  the  slit  ;  but  the  grating  G  reproduces  the  spectrum  and 
enlarges  it,  in  which  case,  however,  not  absolute  position  but  the  direction 
of  rays  is  the  determining  factor. 

40.  Continued.    Reversed   spectra,   etc.  —  The  equation  underlying  the 
greater  number  of  experiments  in  the  work  with  reversed  spectra  is  of  the  form 

(17)  n\  =  2e  cos  5/2 

where  5  is  the  double  angle  of  incidence  at  either  opaque  mirror  and  e  is  the 
effective  distance  apart  of  the  two  mirrors  —  i.e.,  the  distance  between  the 
faces  when  one  is  rotated  180°  about  the  axis  of  symmetry  into  parallelism 
with  the  other.  In  the  present  case,  therefore,  e  corresponds  to  N  in  §  39. 
The  angle  5=02—  0i,  where  02  and  Q\  are  the  angles  of  refraction  of  the  col- 
lecting and  the  dispersing  grating,  respectively,  so  that  sin  8  =  \/D  if  D  is 
the  grating  space.  If  the  silvered  right-angled  reflecting  prism  is  used  for 
alining  the  separated  pencils,  5  =  90°—  61. 
From  the  above  equation  the  change  of  X  per  fringe  is 

d\  _      _  X2  _ 

dn  ~  ~~  e(2  cos  d/2+\(dd/d\)  sin  8/2) 
Since  Xd5/dX=X/£>  cos  0  =  tan  6 

d\  X* 


dn         e(2  cos  5/2+sin  8/2  (tan  02-tan  00) 

This  is  a  maximum  if  e  =  o,  as  the  quantity  in  the  parenthesis  can  not  vanish 
for  very  acute  angles,  such  as  0  and  5  must  be. 

If  5  =  o,  or  Dl=D2>  d\/dn  =  -\2/2e. 

Similarly,  since  e  is  now  the  micrometer  variable  or  coordinate,  and  n 
constant, 

,    ,  d\  2\ 

(20)  —  =  - 

de      e(2+tan  8/2  (tan  02-tan  00) 

from  which  the  similar  conclusions  may  be  drawn  with  regard  to  the  motion 
of  a  given  fringe.  There  is  a  maximum  for  e  =  o.  The  equation,  it  will  be 
seen,  is  quite  cumbersome,  so  that  further  treatment  is  inexpedient.  Never- 
theless equation  (20),  if  0  and  5  are  expressed  in  terms  of  X,  should  admit 
of  integration,  at  least  approximately. 

The  equation  n\  =  2e  cos  (90°—  0)/2  for  reflecting  prisms  needs  special 
treatment,  since  90°  is  not  derived  from  0.  The  coefficients  after  reduction 
become 


dn 
and 

d\     2X(£>+X)     2(£>+X)>/2 

In  both  cases  there  is  a  maximum  for  e  =  o. 


REVERSED  AND   NON-REVERSED   SPECTRA.  81 

Equation  (22)  may  be  integrated,  and  if  C  is  an  experimental  constant, 


There  remains  the  equation  for  crossed  rays  or  achromatic  conditions, 

i8o°-20 

n\  =  ze  cos  —        —  =  2e\/D 

or 

(24)  e  =  nD/2 

in  which  there  should  be  no  motion  of  fringes  throughout  the  spectrum,  but 
for  secondary  reasons. 

It  is  now  possible  to  consider  the  above  results  on  the  increase  of  the  range 
of  displacement  within  which  interferences  are  visible,  with  the  dispersion  of 
the  grating.  In  this  case  the  equations  in  d\/de,  viz,  Nos.  20,  22,  as  well  as  23 
and  24,  may  be  consulted.  Since  sin  0  =  X/D,  all  of  them  involve  the  dis- 
persion i/D,  where  D  is  the  grating  space.  In  the  case  of  No.  24  the  range 
of  displacement  should  be  indefinite,  since  the  locus  of  fringes  is  stationary  in 
the  spectrum.  It  is  found  to  be  exceptionally  large,  but  limited  by  special 
diffraction.  Equation  (20)  is  cumbersome,  but  otherwise  similar  to  equation 
(22),  which  may  be  treated  first.  The  displacement  of  any  given  fringe  in 
wave-length  increases  with/=  i/D,  the  number  of  lines  per  centimeter.  If  a 
fringe  travels  between  any  two  wave-lengths  X  and  X'  and  if  D  is  large  relative 
to  X,  equation  (23)  shows  that  approximately 


The  range  of  displacement  should  therefore  be  roughly  proportional  to  the 
square  root  of  the  dispersion,  and  one  is  not  at  once  at  liberty  to  conclude  that 
the  uniformity  of  wave-trains  is  enhanced  by  dispersion. 

In  fact,  if  D  is  not  large  compared  with  X,  as  in  the  higher  orders  of  dis- 
persion, the  full  equation  must  be  taken.  Unfortunately  the  data  of  Chapter 
I,  §  25,  table  12,  which  are  the  most  complete,  do  not  easily  admit  of  compu- 
tation in  full.  I  have  compared  them  both  with  e  =  e0\/\/(D+\),  in  which 
the  ratios  of  e  observed  and  computed  run  up  with  D,  regularly,  from  i  to  7  ; 
and  with  2^/dX  =  C(2D+X)/(D+X)8/2,  in  which  the  regular  change  of  ratios 
for  the  same  D  (as  D  decreases)  is  again  from  i  to  7.  In  other  words,  the 
observed  values  of  e  varying  with  D  decrease  enormously  faster  than  coeffi- 
cients of  this  type,  as  they  should.  In  view  of  the  equation 


it  follows  that 

de     —  eQ     I 
~         V 


dD~    2 

and  the  comparison  of  the  e  observed  in  table  12  with  this  coefficient  is  there- 
fore crucial. 


82 


THE   INTERFEROMETRY  OF 


To  determine  D  from  table  12  we  have  D=D'/n=i/(cos  6n'd6n/d\), 
where  «  is  the  order  of  the  spectrum  and  d8n/d\  its  dispersive  power,  like  0» 
is  given  in  the  table.  In  this  way  the  data  of  table  15  were  obtained. 

TABLE  15. — Ratio  of  the  range  of  displacement  e  observed  in  table  13  and 
de/dD,  computed. 


Ratio 

Order. 

DXitf 

I  tfe  observed. 

VV(0+A)» 

eoWVV(£+X)3 

0 

1,320 

38 

I5i 

2-5 

i 

335 

200 

980 

2.1 

2 

3 

3 

420 
520 

1  ,800 
2,400 

2-3 
2.2 

4 

142 

58o 

2,690 

2.2 

In  view  of  the  character  of  the  results  for  e,  the  ratio  e/Vx/GD-hA)3,  where 
e  is  the  observed  range  of  displacement,  may  be  considered  constant.  The 
enormous  variation  of  the  range  e  with  the  dispersive  power,  as  observed,  must 
therefore  be  regarded  as  in  keeping  with  the  theory  of  the  phenomenon, 
although  the  computation  is  not  direct.  The  latter  would  require  an  integra- 
tion of  equation  (22)  which  may  be  written 


f" 

-U 


d\ 


2.D+X 
\A(Z?+A)« 
but  the  simpler  comparison  given  was  regarded  adequate. 

Data  bearing  on  equation  (20)  are  given  in  table  n  and  may  after  reduc- 
tion be  written  as  in  table  16  (62  =  19°  30',  D2=  177 X  io"6). 

TABLE  16. 


Range  e 
observed. 

0i 

Z?Xio« 

d 

tan  5/2  X 
(tan  61—  tan  0i) 

0-33 
•52 

2°  36' 

9°  39' 

1,320 
352 

16°  54' 
9°  5i' 

0.0459 
.0156 

The  value  of  the  term  in  the  last  column  is  thus  small  in  comparison  with 
2  and  may  be  neglected,  as  a  first  approximation.    Hence  roughly 


and  the  range  of  displacement  should  be  nearly  independent  of  the  dispersion. 
As  it  is  not,  some  corresponding  principle  must  here  be  active,  and  this  has 
already  been  found  in  the  shift  of  one  illuminated  strip  on  the  collecting  grating 
relative  to  the  other,  when  either  of  the  opaque  mirrors  is  displaced. 

For  the  same  reason  the  effect  produced  by  making  d  =  o,  as  in  §24,  is  not 
marked,  so  far  as  equation  (20)  is  concerned.  If  6  =  o  rigorously,  the  original 
experiment  with  but  a  single  grating  did  in  fact  show  large  ranges  of  displace- 
ment, in  view  of  the  absence  of  sliding. 

We  thus  return  to  the  special  diffraction  already  mentioned  in  §38.  When 
two  spectra  from  the  same  source  coincide,  horizontally  and  vertically, 
throughout  their  extent,  they  will  interfere  at  every  point.  The  interference 


REVERSED  AND   NON-REVERSED   SPECTRA.  83 

will  be  visible  within  a  certain  range  of  path-difference.  If  one  spectrum 
shrinks  longitudinally  on  the  other,  the  strip  carrying  fringes  rapidly  dimin- 
ishes in  breadth;  but  interference  is  still  marked  near  the  transverse  line  of 
coincident  wave-lengths.  If  one  spectrum  is  reversed  with  reference  to  the 
other  on  a  transverse  axis,  the  interferences  are  reduced  to  a  single  nearly 
linear  strip  coincident  with  the  line  of  symmetry.  The  width  of  this  strip 
is  independent  of  the  dispersion  of  the  system.  It  depends  on  the  breadth  of 
colored  region  which  contributes  rays  to  the  strip  in  question.  Hence,  if, 
beginning  with  both  ends  of  the  spectrum,  rays  are  cut  off  except  those  very 
near  the  line  of  symmetry,  the  linear  phenomenon  rapidly  increases  in  size 
until  all  light  is  extinguished.  This  is  what  one  would  expect  from  the 
theory  of  the  diffraction  of  wave-fronts  broad  or  slender,  with  the  generaliza- 
tion as  to  the  rotation  of  fringes  to  which  I  have  already  referred. 

The  association  of  the  two  diffractions  is  well  illustrated  by  the  experiments 
with  inverted  spectra.  Here  the  edge  of  the  reflecting  prism  is  horizontal 
and  normal  to  the  interfering  beams.  When  this  edge  is  moved  normal  to 
itself,  path-difference  only  is  introduced.  To  compensate  a  plate  0.434  cm. 
thick  the  motion  should  be  about  0.243  cm.  The  displacement  found  was 
2^=0.247  cm.,  the  difference  being  referable  to  insufficiently  accurate  dis- 
persive constants.  When  either  of  the  opaque  mirrors  moves,  the  correspond- 
ing beam  slides  along  the  edge  of  the  prism  and  the  displacement  20=0.370 
cm.  of  mirror  was  found,  corresponding  to  the  path-difference  x  =  ze  cos  5/2  = 
0.282  cm.  About  14  per  cent  more  path-difference  is  thus  needed  with  sliding 
(x)  than  without  (zy). 

If  now  the  reflecting  prism  is  turned  90°,  so  that  the  edge  is  vertical,  the 
corresponding  beams  slide  normally  to  or  from  the  edge  of  the  prism  when 
the  opaque  mirrors  are  moved.  The  corresponding  data  were  then  found  to 
be  2^=0.250  cm.,  x  =  0.23 5  cm.  Here  about  6  per  cent  less  path-difference 
is  needed  with  sliding  (x)  than  without  (zy).  The  smaller  effect  in  the  latter 
case  is  to  be  expected,  since  the  two  corresponding  rays  slide  toward  each 
other  in  the  same  plane  and  can  not  pass  through  each  other.  In  the  former 
case  they  pass  in  marked  degree  across  or  through  each  other  and  must  there- 
fore essentially  contribute  to  the  rotation  of  fringes.  But  the  sign  of  the 
effect  is  precisely  the  opposite  to  what  one  would  expect.  Investigations  such 
as  these  and  the  corresponding  question  of  the  width  of  strip  carrying  inter- 
ference fringes  in  case  of  crossed  rays  call  for  apparatus  with  better  optical 
plate,  or  for  more  rigorous  instrumental  adjustment  than  I  have  been  able 
to  utilize  in  the  present  papers.  It  is  best,  therefore,  to  waive  them  for  the 
present,  however  interesting  the  theoretical  results  with  which  they  are 
associated.*  ^ 

*  I  have  since  treated  the  outstanding  difficulties  of  the  text  rigorously  and  the  results 
will  be  given  in  a  future  report. 


CHAPTER  III. 


ELONGATION  OF  METALLIC  TUBES  BY  PRESSURE  AND  THE  MEASUREMENT 
OF  THE  BULK  MODULUS  BY  DISPLACEMENT  INTERFEROMETRY. 

41.  General  method  and  apparatus. — About  25  years  ago*  I  obtained  satis- 
factory results  in  the  measurement  of  pressures  of  the  order  of  1,000  atmos- 
pheres by  the  expansion  of  steel  tubes  of  suitable  thickness.  The  tube  in 
this  case  was  inclosed  in  a  snugly  fitting  glass  tube  filled  with  water  and  the 
volume  expansion  measured  by  an  attached  capillary  tube,  the  system  being 
submerged  in  water  to  obviate  thermal  discrepancies.  The  whole  subject  has 
since  been  transformed  by  the  famous  experiments  of  Prof.  P.  W.  Bridgman, 
and  I  merely  touch  it  here  with  the  purpose  of  testing  the  optic  apparatus 
involved  and  with  a  view  to  the  experiment  explained  in  the  final  paragraph 
of  this  paper.  In  the  present  experiments  I  shall  attempt  to  measure  the 
increase  of  length  of  a  steel  tube  due  to  internal  pressure,  by  the  displacement 
interferometer.  The  experiments  will  lead  to  an  independent  method  for  the 
measurement  of  the  bulk  modulus  (Tait)  and  to  a  procedure  for  studying  the 
thermodynamics  of  the  adiabatic  expansion  of  liquids. 


The  interferometer  used  was  of  the  linear  type  (fig.  54).  Here  L  is  a  weak 
lens,  about  2  meters  in  focal  distance  and  12  cm.  in  diameter,  concentrating 
a  beam  of  sunlight  on  the  slit  5.  c  is  the  objective  of  the  collimator,  being 
a  spectacle  lens  of  about  i  meter  focal  distance.  It  is  particularly  advanta- 
geous to  have  rays  of  slight  obliquity  here  if  a  brilliant  and  wide  spectrum  is 
to  be  seen  in  the  telescope  at  T.  H  is  the  half-silvered  plate  of  the  inter- 
ferometer, N  and  M  (on  a  micrometer)  are  the  opaque  mirrors,  each  about  2 
meters  from  H.  The  rays  reaching  the  telescope  T  would  therefore  show 
two  white  slit  images  from  N  and  from  M,  which  are  to  be  placed  in  coin- 
cidence both  horizontally  and  vertically  by  the  adjustment-screws  on  M  and 

*  Phil.  Mag.  (5),  xxx,  p.  338,  1890;  Bull.  U.  S.  Geol.  Sur.,  No.  96,  1892;  cf.  Am.  Acad. 
Arts  and  Sciences,  xxv,  p.  93,  1890;  for  effect  of  pressure  on  electrical  conductivity  of 
liquids,  see  Am.  Journal,  XL,  p.  219,  1890,  and  on  the  mercury  pressure-gage,  Am.  Journal, 
p.  96,  XLV,  1893. 

84 


INTERFEROMETRY   OF   SPECTRA.  85 

N.  To  bring  out  the  interferences,  a  direct-vision  prism  grating  g  is  placed 
in  front  of  the  objective  of  T,  whereupon,  when  the  path-difference  HM,  HN 
is  annulled,  magnificent  ellipses  may  be  seen  in  the  bright  spectrum  in  the 
field  of  the  telescope. 

The  steel  or  other  tube  whose  elongation  under  pressure  is  to  be  measured 
is  shown  diagrammatically  at  1 1'.  The  end  t',  moreover,  is  closed  by  a  tinned- 
steel  plug-screw,  while  t  communicates,  by  aid  of  a  thick-walled  ^-inch  tube 
p  of  small  bore,  with  the  screw-compressor  P  described  in  my  earlier  papers 
(/.  c.}.  It  is  here  that  the  thick  hydrocarbon  oil  is  forced  into  the  tube  1 1' 
and  the  pressure  measured  by  a  Bourdon  pressure-gage  G,  reading  in  steps 
of  10  atmospheres  to  1,000  atmospheres. 

The  parts  of  the  interferometer  are  attached  directly  or  indirectly  to  a 
brick  pier  in  the  laboratory.  M  is  separately  so  attached;  so  is  also  the  end 
/  of  the  steel  tube  by  the  bracket  at  B,  this  being  fixed  rigidly.  The  other 
end,  t',  which  must  be  free  to  expand,  is  to  be  supported  on  knife-edges  or 
rollers  of  a  vertical  pendulum  hanger  y,  the  supports  of  which  are  in  turn 
rigidly  fixed  to  the  pier.  This  will  presently  be  further  described.  It  was 
found  that  long  vertical  wires,  supporting  intermediate  parts  of  the  tube, 
were  desirable  and  quite  as  good  as  more  complicated  arrangements. 

With  the  tube  1 1'  thus  fixed  except  as  to  linear  expansion  toward  the 
right,  the  mirror  N  is  clamped  by  a  horizontal  lateral  arm  at  the  end  t',  and 
the  half-silvered  plate  H  by  a  similar  arm  on  the  other  side,  at  the  end  t. 
Thus  the  length  HN  varies  with  the  pressure  and  the  increment  is  compen- 
sated at  M  by  bringing  the  center  of  ellipses  back  to  the  D  lines  in  the  field 
of  the  telescope.  Accessories  like  water-jackets,  etc.,  are  left  out  of  the  fig- 
ure for  clearness  and  will  not  be  used  in  these  experiments. 

To  obviate  friction,  the  end  t'  of  the  tube  1 1'  was  suspended  -from  a  rec- 
tangular yoke  or  pendulum  consisting  of  two  vertical  rods  y  and  y',  figure 
55,  and  horizontal  smooth  round  brass  cross-rods  r  and  r'.  The  latter  roll  on 
the  round  horizontal  rods  a  and  b  suitably  anchored  at  the  same  level  in  the 
pier.  The  rod  r'  supported  the  free  end  i'  of  the  pressure  tube. 

To  counteract  vibrations  the  rod  y  carries  a  bell-shaped  damper,  d,  below, 
submerged  in  oil  in  the  cup  c.  The  middle  of  the  tube  i'  is  similarly  damped 
at  its  center  by  a  bell-shaped  damper  in  oil  (not  shown),  against  lateral  and 
vertical  vibrations. 

42.  Remarks  on  the  displacement  interferometer.— As  a  rule  the  ellipses  are 
not  seen  at  their  best  in  the  principal  focus.  The  ocular  must  be  drawn  out 
somewhat  to  focus  them  sharply;  but  usually  the  sodium  lines  are  still  visible 
for  guidance.  No  doubt  this  is  due  to  the  fact  that  ordinary  glass  plate  was 
employed  at  M ,  N,  and  H,  figure  54,  or  that  H  was  not  optically  plane  parallel. 
Moreover,  as  M  is  displaced,  the  focus  of  fringes  changes;  but  as  the  centers 
of  ellipses  are  used  in  measurement  this  is  no  particular  disadvantage. 

A  few  trials  were  made  with  lens  compensators,  but  the  available  combi- 
nations reduced  the  ellipses  to  horizontal  sharp  spindles  or  lines,  without 


86  THE   INTERFEROMETRY  OF 

passing  into  the  hyperbolic  patterns.  However,  a  pair  of  plates  of  about  the 
same  thickness  (0.7  cm.),  but  respectively  of  crown  and  flint  glass,  when  in- 
serted in  the  M  and  N  beams,  gave  good  hyperbolas.*  It  was  interesting 
to  notice  that  in  the  principal  focus  the 
fringe  pattern  was  nearly  circular  or 
roundish  elliptic,  but  the  ellipses  were  not 
strong.  On  drawing  the  ocular  out,  these 
ellipses  underwent  continuous  deforma- 
tion, passing  through  horizontal  bands 
finally  into  sharp  and  strong  hyperbolic 


d 


forms.  The  fringes  obtained  with  the  ^  ** 
differently  dispersing  plates  must  therefore  be  referred  to  space  loci  in  which, 
on  passing  along  the  axis,  an  elliptic  cone  becomes  linear  at  the  apex  and 
symmetrically  hyperbolic  beyond.  Thus  in  figure  56  the  sections  of  the  cone 
at  a,  b,  c  show  fringes  of  the  form  d,  e,  f. 

43.  Observations.  Thick  steel  tube. — The  seamless  steel  tube  in  my  pos- 
session had  the  following  dimensions:  Length  between  mirrors,  160.8  cm.; 
diameter  inside,  0.515  cm.;  diameter  outside,  1.278  cm.  Hence  the  inside 
cross-section  is  0.208  square  cm.  and  the  cross-section  of  steel  1.075  sq.  cm. 
Such  a  tube  is  therefore  rather  adapted  for  measuring  very  high  pressures, 
whereas  the  following  work  will  not  go  beyond  1,000  atmospheres;  but  it 
was  admitted  for  trial.  If  the  case  is  treated  as  simple  traction  with  Young's 
modulus  taken  as  2. 14X10 u,  the  elongation  should  be  about  i4Xio'6 
cm.  per  atmosphere,  equivalent  to  a  little  less  than  half  an  interference 
ring.  But  this  is  obviously  too  large,  as  the  tube  expands  in  all  directions 
by  internal  pressure.  The  case  has  been  treated  by  Tait  and  will  presently 
be  referred  to. 

The  experiments  made  proceeded  with  unexpected  smoothness  from  the 
outset,  barring  the  tremors  of  the  laboratory,  which  made  it  difficult  to  set 
the  ellipses.  Comparing  the  steel  gage  with  a  standard  Bourdon  gage  read- 
ing in  steps  of  10  atmospheres  to  1,000  atmospheres,  the  first  six  series  of 
experiments  showed  an  elongation  AL  of  tube  per  atmosphere  of 

io6AL  =  7.s,        8.0,        7.0,        8.0,        7.0,         7.5  cm. 
In  the  last  series  the  individual  results  were: 

Pressure 100    200    300    400    500  500    400    300    200    100  atmosph. 

Micrometer  at 93      87      79      71      65  65      73      81       89      96  cm./io« 

lo'AL  =  7.4 X  icr*  cm.  io8A£  =  7-7  X  icr6  cm. 

The  data  for  decreasing  pressures  do  not  return  into  the  preceding  values 
for  increasing  pressures  for  incidental  reasons  which  need  not  be  discussed, 
but  the  mean  rates  are  not  very  different.  The  discrepancies  are  probably 
in  the  setting  of  the  Bourdon  gage.  They  are  not  due  to  temperature  here, 

*  Later  experiments  showed  that  the  flint  plate  was  slightly  curved. 


REVERSED  AND   NON-REVERSED   SPECTRA.  87 

even  though  the  tube  was  not  jacketed  with  water.  The  data,  as  was  to  be 
expected,  show  an  expansion  less  than  was  computed  for  the  case  of  traction, 
being  only  about  half  as  large,  or  equivalent  to  about  a  quarter  of  an  inter- 
ference ring  per  atmosphere.  An  arc  lamp  was  used  as  a  source  of  light  and 
its  flickering  was  very  annoying. 
Tait*  in  the  Challenger  reports  justifies  the  expression 


_ 

dx        ai2  -  ao2    36 

where  d%/dx  is  the  elongation  due  to  the  internal  pressure  TI,  in  case  of  a 
tube  of  sectional  diameters  oo  and  ai  and  bulk  modulus  k.  This  merely  re- 
places Young's  modulus  by  the  linear  equivalent  of  the  bulk  modulus.  Hence 
for  the -tube  of  length  x  =  160.8  cm.  and  00=0.515  cm.,  01=1.278  cm.,  if  for 
tool  steel  k  =  i.84X  io6  and  pressure  is  given  in  atmospheres  (io6  dynes/cm.2) 
the  elongation  per  atmosphere  is 

160.8X0.265 

*  cm. 


1.368X5-  52Xio6 

This  is  smaller  than^the  value  found,  probably  because  the  tube  is  made  of 
mild  steel.    If  k  be  computed  for  the  tube,  since  the  elongation  per  atmosphere 

160.8X0.265 


which  is  about  the  order  of  value  given  by  Everett^  for  wrought  iron.    Voigt 
gives  r.46Xio6  for  steel  (see  Landolt  and  Boernstein's  Tables,  1905,  p.  45). 

44.  Further  experiments.  —  The  washer  of  the  screw  of  the  compressor,  in 
the  default  of  marine  glue,  was  a  perforated  disk  of  pitch.  This  proved  to 
be  quite  inadmissible  for  further  work.  The  behavior  of  pitch  was  very 
peculiar  and  in  itself  interesting.  The  perforated  disk  was  found  to  adhere 
without  slip  to  the  screw  at  the  inner  edge  r,  figure  57,  and 
to  the  wall  of  the  stuffing-box  at  its  outer  edge  R.  On 
turning  the  screw  a  smaller  coaxial  disk  r'  of  pitch  broke 
out  of  the  larger  disk,  and  the  turning  proceeded  on  this 
surface,  r',  without  leakage.  It  was  found  nearly  impos- 
sible to  force  the  screw  with  the  adhering  pitch  into  the 
socket  without  serious  injury  to  the  screw.  The  sharp 
edges  of  the  threads  were  in  fact  planed  off  flat  and  it  was  only  by  leaving 
fore-and-aft  room  for  play  in  the  stuffing-box  that  the  compressor  could  be 
used  at  all.  The  part  of  the  screw  turning  in  pitch  was  ruined.  _  ^ 

*  Tait:  Challenger  Reports,  n,  1882,  App.  A,  p.  26. 

t  See  Everett  tables,  1879,  p.  S3,  containing  original  experiments  of  the  author. 


THE   INTERFEROMETRY  OF 


The  pitch  washer  was  therefore  removed  and  replaced  by  one  of  tallow 
slightly  hardened  with  a  little  resin  or  wax,  the  two  being  melted  together. 
This  functioned  perfectly  within  1,000  atmospheres  and  was  easily  inserted 
in  parts  which  could  then  be  molded  into  a  disk  within  the  stuffing-box  on 
forcing  the  gland  into  it. 

The  measurements  given  in  table  17  were  made  in  steps  of  100  atmos- 
pheres. AL  denotes  the  elongation  per  atmosphere. 

TABLE  17. 


Series. 

Pressure  range. 

io"AL 
(pressure 
increasing)  . 

lo'AL 
(pressure 
decreasing). 

I 

9 

10 

lootosooatm. 

100       600 

100     700 

100       800 

8.3  cm. 

9-2 

7-9 
7-9 

6.5  cm. 
6.1 

6.2 

7-1 

An  example  of  the  individual  results  may  be  given  in  case  of  series  9 : 

Pressure 100  200  300  400  500  600  700  700  600  500  400  300  200  looatm. 

Micrometer  reading  i. 01   9.4  8.6    7.9  7.0  6.2  5.5    5.4  6.3  7.0    7.7  8.5    9.1   g.Scm./io3 

The  elongation  here  is  always  greater  when  pressures  increase,  although 
time  is  allowed  for  dissipation  of  temperature,  than  when  they  decrease. 
Four  reasons  may  be  assigned  for  this  result:  (i)  the  temperature  increase 
on  increasing  pressure  and  vice  versa;  (2)  permanent  set  imparted  to  the 
tube;  (3)  elastic  warping  of  the  tube  owing  to  the  end-thrust  of  internal 
pressures  and  consequent  disadjustment  of  the  interferometer;  (4)  vis- 
cosity of  steel.  Probably  all  of  these  discrepancies  are  present.  That  there 
was  set  I  infer  from  the  gradual  displacement  of  the  reading  of  the  interfer- 
ometer for  100  atmospheres  at  the  beginning  and  end  of  a  series,  though  this 
may  be  due  to  new  adjustments.  The  incidental  displacements  are  particu- 
larly shown  in  the  values  of  AL  when  pressure  increases  and  are  specially 
marked  in  series  7  and  8.  They  are  also  apparent  in  the  change  of  form  of 
the  ellipses.  As  sunlight  was  used  in  the  above  work  the  annoyances  of  a 
flickering  arc  do  not  occur.  The  ellipses  were  not  centered. 

To  obtain  some  notion  of  the  relation  of  these  discrepancies  we  may  pro- 
ceed as  follows :  The  difference  between  the  elongation  per  atmosphere  dur- 
ing the  phases  of  increasing  and  decreasing  pressures  in  the  four  series 
given  is,  respectively,  1.8,  3.1,  1.7,  0.8  cm./io6,  or  i.SXio"6  cm.  per  at- 
mosphere of  compression.  For  a  tube-length  of  160.8  cm.  and  a  coefficient 
of  expansion  12  X  io"6  this  is  equivalent  to  a  rise  of  temperature  9.3  X  io"4,  or, 
roughly,  io"3  C.  per  atmosphere  of  compression. 

Supposing  the  compressibility  of  the  oil  to  be  cfo=iooXio~6  per  atmos- 
phere per  cubic  centimeter  and  the  mean  pressure  £  =  500  atmospheres,  the 
work  done  is  pdv  or 

5ooXio6XiooXio"6  =  5Xio4  ergs  per   atmosphere  per  cubic  centimeter  at 
500  atmospheres 


REVERSED  AND   NON-REVERSED   SPECTRA.  89 

Since  the  preceding  datum  corresponds  to  both  increasing  and  decreasing 
pressures,  the  work  done  must  be  reckoned  per  2  atmospheres  or  it  will  be 
IOB  ergs.  Taking  the  specific  heat  of  oil  as  0.5  and  the  mechanical  equivalent 
as  42  X  io6,  the  rise  of  temperature  of  the  oil  should  be 


io5 


5  Xio-3°C.,  nearly 


Hence  the  residual  temperature  discrepancy  found,  o.ooi  C.,  would  be  but 
one-fifth  of  the  full  temperature  discrepancy  to  be  anticipated — i.e.,  four- 
fifths  of  the  heat  would  have  dissipated  by  conduction,  etc.,  during  the  wait- 
ing between  successive  compressions. 

The  residual  temperature  is  thus  adequate  to  account  for  the  full  discrep- 
ancy, and  if  a  tube  of  this  kind  is  to  be  used  as  a  pressure-gage,  the  tube  should 
be  made  of  "invar"  or  other  metal  without  thermal  expansion.  True,  a 
water-jacket  surrounding  the  tube  would  improve  the  apparatus,  but  the 
thermal  increments  in  question  are  so  small  that  the  device  would  not  be 
trustworthy.  If,  however,  a  temperature  discrepancy  is  shown  by  the  optic 
gage  it  should  also  be  shown  by  the  more  sensitive  Bourdon  gage,  which  is 
not  the  case.  Thus  a  residual  effect  of  temperature  is  improbable.  The 
optical  difficulties  are  slight  and  could  be  overcome  by  suspending  the  yoke, 
which  in  figure  55  rolls  loosely  on  the  cylinder  a,  b,  tr,  from  steel  pivots  bearing 
on  jeweled  cups.  Elastic  and  particularly  slow  viscous  yieldings  to  persistent 
pressure  are  thus  the  probable  reason  for  the  errors.  This  also  accounts  for 
the  displacement  of  the  fiducial  reading. 

Two  further  experiments  were  now  made  in  which  the  ellipses  were  centered 
before  each  observation  (table  18).  In  the  second  set  (series  12)  the  tube 
was  attached  to  the  yoke  and  the  latter  to  the  hangers  by  soft  adhesive  wax, 
applied  in  the  molten  state.  This  proved  quite  adequate. 


TABLE  18. 


Series. 

Range. 

iosAZ< 
(pressure 
increasing). 

io*AL 
(pressure 
decreasing). 

II 

12 

iooto6ooatm. 
100     600 

7.2 

7-7 

7.2 

7-7 

The  zeros  were  regained  and  the  results  were  marked  improvement  on  the 
preceding  series.  The  same  mean  elongation  is  found  for  increasing  and  de- 
creasing pressure;  but  the  values  are  not  identical  in  the  two  series.  The 
following  data  give  the  details  of  series  1 1 : 


Pressure 100 

Micrometer  reading     50 


43 


300 
34 


400 
28 


500 
20 


600 
15 


500 

21 


400 
28 


300 
36 


200 

43 


100  atm. 
50  cm./io4 


A  few  units  in  the  cm./io4  place  are  thus  uncertain, 
in  figure  58. 


The  graph  is  shown 


90 


THE  INTERFEROMETRY  OF 


45.  Brass  tube.— A  somewhat  thinner  seamless  tube  of  soft  brass  was  next 
tested  within  600  atmospheres.  The  dimensions  were:  length,  161  cm.; 
diameter  within,  0.485  cm. ;  diameter  outside,  0.960  cm.  Hence 


161X0.235 


(0.9216  —  0.2352)3^ 


18.36 
AL 


The  ellipses  were  centered  throughout  and  the  yoke  was  given  additional 
stability  by  soft  adhesive  wax,  as  above.  The  tube  showed  extraordinary 
variability,  but  during  the  trials  under  increasing  and  decreasing  pressure 
and  in  the  lapse  of  time  the  viscous  changes  somewhat  subsided,  as  will  be 
seen  from  table  19. 

TABLE  19. 


Series. 

Range  of 
pressures. 

io«AL 
(pressure 
increasing). 

io6AL 
(pressure 
decreasing)  . 

i 

2 

3 

4 

lootosooatm. 
100     500 
100      500 

100       600 

13.6  cm. 
13-5 
13-7 

15.6  cm. 
16.5 
17.0 
16.7 

Mean. 

13.6 

16.7 

Disregarding  the  first  series,  which  was  preliminary,  the  remaining  data 
are  consistent.  Hence  k  from  the  mean  io6AL=is.2  is  relative  to  atmos- 
pheres 

,18.36 
15.2 


I.2lXlO« 


Voigt's  value  for  brass  is  but  0.61  X  io6,  i.e.,  but  half  this.  Throughout  these 
experiments  the  reading  for  100  atmospheres  wandered  continually,  creeping 
over  7  X  io'3  cm.  during  the  time  interval  of  the  experiments.  The  following 
individual  data  show  this  for  the  fourth  series: 

Pressure 100   200   300   400   500   600   600   500   400   300   200    100  atm. 

Micrometer  reading  320   301    290   274   263    250    252    270   290   305    321    336  cm./io* 

One  would  naturally  refer  this  to  the  viscosity  of  the  brass  tube,  but,  curiously 
enough,  the  march  is  a  contraction.  Apparently  the  tube  continually  con- 
tracts in  the  lapse  of  time  under  internal  pressure.  The  contraction  occurs, 
however,  for  the  case  of  a  tube  which  was  not  rigorously  straight. 

Optically,  apart  from  the  tremor  of  the  laboratory,  one  would  have  no 
fault  to  find  with  the  measurements,  allowing  a  micrometer  accuracy  of  a 
few  io-4  cm.  Interference  rings  could  easily  have  been  utilized,  but  this 
would  have  required  two  observers. 

Two  more  series  of  experiments  were  made  with  the  brass  tube  (table  20), 
in  one  (5)  of  which  it  was  supported  only  at  the  ends  with  its  original  curva- 


REVERSED  AND   NON-REVERSED   SPECTRA. 


91 


ture  convex  upward;  in  the  sixth  series  the  tube  was  additionally  supported 
in  the  middle  on  a  large  pendulum-like  wire  or  hanger. 


TABLE  20. 


Series. 

Range  of 
pressure. 

io6AL 
(pressure 
increasing)  . 

io»AL 
(pressure 
decreasing). 

5  (free)  
6  (supported)... 

iooto6ooatm. 
100      500 

12.3  cm. 
13-6 

15.2  cm. 
15-2 

The  individual  results  are  shown  in  figures  59  and  60.  It  will  be  noticed 
that  an  error  was  introduced  when  pressures  passed  from  the  increasing  to 
the  decreasing  phase  at  the  highest  pressure,  and  this  was  particularly  the 
case  when  a  slight  leak  developed.  If  the  highest  pressure  is  omitted,  io6AL 
=  14.7,  both  for  increasing  and  decreasing  pressures  in  the  case  of  the  sus- 
pended tube.  Hence  k  =  i.25oX  io6,  not  differing  essentially  from  the  above. 
The  unsupported  tube  is  highly  subject  to  viscosity. 


•WO    SCO 

The  large  effect  resulting  from  viscosity  is  also  shown  in  the  initial  and 
final  readings  (100  atmospheres)  in  figures  59  and  60.  It  is  in  both  cases 
again  an  apparent  contraction  and  is  present  even  in  the  supported  tube. 
The  viscous  effect  in  the  lapse  of  time  is  directly  given  in  the  following  inde- 
pendent measurements  of  apparent  contraction.  The  tube  was  kept  charged 
with  an  internal  pressure  of  100  atmospheres. 

Time 9h3Om  ioh  5°  I2h  I2» 

Length,  L 0.0173  0.0183 

It  is  difficult  to  understand,  however,  how  anything  of  the  nature  of  vis- 
cous longitudinal  contraction  can  occur  under  internal  pressure.  One  might 


92 


THE   INTERFEROMETRY   OF 


suppose  that  the  cylindrical  tube  as  a  whole  is  gradually  progressing  toward 
an  ultimate  spherical  form,  but  this  seems  far-fetched.  It  is  more  reasonable 
to  suppose  that  the  viscosity  is  simply  flexural.  The  tube  is  curved  slightly 
convex  upward  and  therefore  the  end  mirrors  N  and  H,  figure  54,  rotate  con- 
tinually towards  each  other  on  a  horizontal  axis,  under  the  end-thrusts  of  the 
internal  pressure.  The  component  beams  and  HNH  and  HMH  are  thereby 
each  modified  in  length.  Though  it  is  difficult  to  specify  why  the  former 
should  be  shortened  relative  to  the  latter,  such  a  result  is  easily  conceived. 

46.  Thin  steel  tube. — The  steel  tube  in  §§43  and  44  was  adapted  for  high 
pressures  only,  showing  but  0.2  interference  ring  per  atmosphere.  In  con- 
trast with  this  a  thin  steel  tube  was  now  inserted,  adapted  for  lower  pressures. 
This  was  more  sensitive  than  the  Bourdon  gage,  the  other  tube  being  on  the 
whole  less  so.  The  dimensions  were:  length,  L=i6i  cm.;  diameter  inside, 
00  =  0.799  cm.;  diameter  outside,  01  =  0.951  cm.  The  outside  diameter  was 
calipered.  A  short  length  was  then  cut  off  and  slit  open  and  the  wall  thick- 
ness similarly  found  (0.076  cm.).  The  tube  was  not  quite  straight.  Snugly 
fitting  brass  plugs  were  carefully  soldered  into  the  ends,  and  these  were  then 
tapped  to  receive  the  tubes  conveying  pressure. 

The  observations  shown  in  table  21  were  recorded,  the  steps  of  pressure 
being  50  atmospheres  in  the  first  two  and  100  atmospheres  in  the  last  two 

series. 

TABLE  21. 


Series. 

Pressure  range. 

io6AL 
(pressure 
increasing)  . 

io6A£ 
(pressure 
decreasing). 

i 

2 

3 

4 

5oto20oatm. 
50     300 
100     400 
ioo     400 

89     cm. 
95-3 
97-3 
95-2 

100.1  cm. 
95-5 

100.0 

100.7 

At  400  atmospheres  the  tube  developed  a  slight  leak  at  the  ends.  At  750 
atmospheres  one  of  the  soldered  end-plugs  was  blown  out.  It  is  remarkable 
that  the  plugs  held  so  well. 

An  example  of  the  individual  data  may  be  given  for  the  second  series.  In 
figure  6 1  these  data  are  shown,  positively. 

Pressure 50    ioo    150   200   250   300   300   250   200    150    ioo     50  atm. 

Micrometer  reading  260    210    164    115      67      23      24      67    120    163    212    266  cm./io* 

Very  little  viscosity  is,  therefore,  in  evidence,  but  there  is  some  displace- 
ment or  irregularity,  probably  in  the  reading  of  the  Bourdon  gage. 

Pressure  increments  and  decrements  slightly  rotated  the  mirrors  in  opposite 
directions  around  both  a  vertical  and  a  horizontal  axis.  These  were  compen- 
sated by  adjustment  at  the  mirror  M  before  each  reading.  As  the  mirrors 
inclined  towards  each  other  for  pressure  increments  the  tube  must  have  been 
slightly  convex  upward,  and  therefore  successively  straightened  as  pressures 


REVERSED  AND   NON-REVERSED   SPECTRA.  93 

increased.    The  mean  elongation  per  atmosphere  is  io6AL  =  9;  cm.  and  the 
bulk  modulus  may  be  computed  as 

oo2        L  0.6384X161 

~ 


a2i-a20  3AL      (0.9044-0.6384)  X3XQ7  X 

In  spite  of  the  large  difference  of  dimensions,  this  datum  is  of  the  same  order 
of  value  as  the  above  (k=  i.sgXio6)  for  the  thick  tube,  particularly  as  the 
present  AL,  from  the  occurrence  of  flexure,  is  probably  slightly  large. 

A  tube  of  this  kind  with  well-sealed  ends  (brazed  probably),  quite  straight, 
and  supported  at  different  points  of  its  length  by  wire  pendula,  should  make 
a  good  pressure-gage  within  at  least  i  ,000  atmospheres.  An  individual  read- 
ing about  id-4  cm.  per  atmosphere  or  over  3  interference  rings  would  be  uni- 
formly available  throughout. 

47.  Conclusion.  Thermodynamic  application.  —  The  data  given  show  that 
an  independent  method  of  measuring  the  bulk  modulus  of  metals  is  quite 
within  the  province  of  the  displacement  interferometer.  The  annoyances 
encountered,  resulting  from  the  viscosity  of  the  metal  or  from  warping,  may 
be  considered  eliminated  in  the  mean  of  the  pressure-increasing  and  pressure- 
decreasing  phase  of  the  experiment.  The  tubes  should  be  supported  at  vari- 
ous points  along  their  length.  Even  the  temperature  discrepancy,  if  sufficient 
time  is  allowed  between  the  successive  steps  of  pressure,  seems  not  to  be  of 
serious  effect  on  the  mean  data. 

For  the  measurement  of  pressure,  how- 
ever, the  device  is  promising.  In  such  a 
case  the  tube  section  should  be  chosen  to 
correspond  with  the  pressures  to  be  meas- 
ured. Within  1,000  atmospheres  a  steel 
tube  about  i  cm.  in  diameter,  with  walls 
about  0.75  mm.  thick,  gave  fair  results, 
showing  the  evanescence  of  about  3  inter- 

ference rings  per  atmosphere.  Such  a  tube  must  be  rigorously  straight,  well 
supported,  and  if  possible  of  non-expanding  (temperature)  steel. 

It  is  interesting  to  consider  the  case  of  the  adiabatic  expansion  of  liquids 
in  relation  to  such  a  gage.  The  available  thermodynamic  equation  is 


where  A  0  is  the  temperature  increment  corresponding  to  the  adiabatic  com- 
pression Ap  at  the  temperature  0,  in  case  of  a  liquid  whose  coefficient  of  ex- 
pansion is  a,  density  p,  and  specific  heat  Cf.,  /is  the  mechanical  equivalent 
of  heat.  In  an  apparatus  like  figure  62,  in  which  P  is  the  screw  compressor 
(with  tinned  or  waxed  screw  5)  filled  with  the  liquid  in  question,  G  the  Bour- 
don gage,  pressures  may  be  suddenly  applied  without  leakage  by  turning  the 


94  INTERFEROMETRY  OF   SPECTRA. 

screw  5.  These  are  also  measurable  at  the  interferometer  gage  g,  H  being 
the  half-silvered  and  TV  an  opaque  mirror,  as  in  figure  54.  The  subsidence 
of  pressure  due  to  cooling  adiabatic  compression  is,  however,  also  measur- 
able at  g  in  terms  of  the  delaying  pressure.  For  we  should  have  (v  denoting 
volume,  k  the  bulk  modulus) 

Av  Av    8p 

—  =  «A0; =~ 

v  v      k 

Sp 
or,  apart  from  signs,  A0=  — ,  where  5p  is  the  subsidence  of  pressure  due  to  the 

cooling  A0,  after  adiabatic  compression.  Hence  the  original  equation  takes 
the  form 

_kaze     Ap 

p~7p~  *P 

an  equation  for  measuring  the  specific  heat  of  constant  pressure  of  the  liquid 

2p-\-Ap 
at  temperature  0  and  pressure  .    The  observations  thus  consist  in 

2 

measuring  Ap  (in  displacement)  and  8p  in  interference  rings,  both  at  the 
gage  g. 

One  may  estimate  the  value  of  dp  per  atmosphere  of  Ap,  for  alcohol,  by 
way  of  illustration.  Here  in  c.g.s.  units, 

fc=i.2iXio10;  a=i.iXio'3;  0=20°;  0  =  0.79;  6^=0.58 
Hence 

i.2iXio10X(i.iXio-3)2X2oXio«  dynes 

*"  42XxoeXo.79Xo.s8  ^^''^nT* 

or 

6^  =  0.015  atm.  per  atm.  of  Ap 

Hence,  if  8p  =  2oo  atmospheres,  the  above  gage  would  show  about  10  rings 
for  dp.  Similarly  a  pressure  dropping  adiabatically  from  1,000  atmospheres 
would  show  about  50  rings,  residually,  after  closing. 

The  present  research  was  planned  to  be  pursued  at  much  greater  length; 
but  owing  to  the  annoyances  of  a  quivering  pier,  which  are  particularly 
marked  during  the  term  weeks,  and  to  the  injury  of  the  screw  before  the  tal- 
low washer  was  inserted,  it  was  thought  wise  to  discontinue  it  at  present. 
If  the  micrometer  is  to  be  set  to  io-4  cm.,  the  ellipses  must  be  reasonably 
quiet,  and  in  case  of  long-distance  interferometry  such  a  condition  can  be 
realized  only  during  the  summer  months. 


CHAPTER  IV. 


REFRACTIYTTY  DETERMINED   IRRESPECTIVE  OF  FORM  BY  DISPLACEMENT 
INTERFEROMETRY. 

48.  Introductory. — Some  time  ago*  I  made  a  number  of  experiments  on 
the  use  of  curvilinear  compensators  in  connection  with  the  displacement 
interferometer.    It  is  obvious  that  the  curvature  in  such  a  case  must  be  very 
small,  so  that  single  lenses  for  the  purpose  are  not  easily  obtained.  The  use 
of  a  doublet  of  two  lenses  of  the  same  glass,  but  respectively  convex  and  con- 
cave, meets  the  case  fairly  well,  the  necessary  refracting  power  being  received 
by  spacing  the  doublet.    Lenses  of  about  i  diopter  each  gave  the  best  re- 
sults, bringing  out  fringes  of  quasi-elliptic  and  hyperbolic  symmetry  in  great 
variety. 

Later  it  appeared  as  if  plates  of  different  varieties  of  glass,  as  for  instance 
crown  and  flint,  if  placed  in  the  two  component  beams  MH,  NH,  figure  54, 
would  produce  the  same  phenomena.  The  flint  plate  used,  however,  proved 
to  be  inadequately  plane,  so  that  the  result  is  in  doubt. 

More  recently  I  have  endeavored  to  secure  similar  results  by  submerging 
the  lens  (convex  or  concave)  in  a  liquid  of  about  the  same  index  of  refraction. 
This  method  would  seem  to  be  interesting  in  other  respects,  for  it  is  probable 
that  the  index  of  the  solid  may  be  determined  in  this  way  irrespective  of 
form.t  If,  for  instance,  the  liquid  and  the  solid  have  the  same  index,  one 
would  be  tempted  to  infer  that  the  latter  may  be  removed  or  inserted  with- 
out displacing  the  center  of  ellipses  at  the  particular  wave-length  under  con- 
sideration. The  index  of  the  liquid  in  place  is  then  determinable  by  the 
interferometer  to  a  few  units  in  the  fourth  place. 

If  experiments  of  the  present  kind  are  to  be  accurate,  it  is  obvious  that 
the  walls  and  cavity  of  the  trough  in  which  the  lenses  are  to  be  submerged 
must  be  optically  plane  parallel;  otherwise  some  compensating  adjustment 
must  be  made  at  the  opaque  mirrors  of  the  interferometer,  and  for  this  no 
adequate  allowance  is  at  hand.  It  did  not,  however,  seem  worth  while  to 
provide  expensive  apparatus  before  the  method  had  been  worked  out  in  de- 
tail. Accordingly  the  present  experiments  were  conducted  with  troughs  of 
ordinary  plate-glass  put  together  by  myself,  and  little  attention  will  be  given 
to  absolute  values  of  index  of  refraction,  as  such. 

49.  Preliminary  experiments.— The  first  experiments  were  made  on  a  large 
linear  interferometer  (see  fig.  54)  with  distances  of  nearly  2  meters  between 

*  Carnegie  Inst.  Wash.  Pub.  No.  249, 1916,  chapter  ix;c/.  Amer.  Journ.  Science,  XL, 

PP  t2MrT3R8W9Cheshire  (Phil.  Mag.,  xxxn,  1916,  pp.  409-420)  has  recently  used  Tc-pler's 

method  for  the  same  purpose  with  marked  success. 


96 


THE   INTERFEROMETRY  OF 


the  mirrors.  The  rays  in  such  a  case  are  all  very  nearly  parallel.  Sunlight, 
arc  light,  and  the  Nernst  filament  were  each  available  for  illumination.  If 
the  latter  is  used,  the  adjustment  must  be  made  by  aid  of  the  two  white  slit 
images,  which  are  to  coincide  horizontally  and  vertically.  Otherwise  the 
sodium  lines  are  available.  With  a  very  long  collimator  (2  meters)  and  a 
wide  single-lens  objective  (10  cm.  or  more),  the  Nernst  filament  may  be  used 
directly  in  place  of  the  slit.  If  the  beam  passing  the  objective  is  not  wider 
than  i  cm.  (opaque  slotted  screen),  very  perfect  ellipses  may  be  obtained. 
On  inserting  the  trough  with  a  thickness  of  1.3  cm.  of  CS2  solution  normally 
into  the  MH  beam,  the  original  very  large  ellipses  were  reduced  in  size  and 
rounded  as  usual  to  smaller  circles.  Submerging  a  convex  lens  (i  diopter) 
into  the  liquid  until  the  beam  passed  symmetrically  through  it  changed 
these  circles  to  very  long  horizontal  spindles.  A  concave  lens  similarly  pro- 
duced horizontally  very  eccentric  hyperbolae.  With  water  in  the  trough,  only 
the  convex  lens  showed  observable  fringes,  these  being  very  long,  practically 
linear  horizontal  spindles.  All  these  fringes  lie  considerably  in  front  of  the 
principal  focal  plane  of  the  telescope  (fig.  54,  T),  and  the  abnormal  forms  are 
necessarily  relatively  faint.  They  change  in  shape  and  intensity  with  the 
focal  plane  observed. 


64 


On  mixing  CSz  with  kerosene  (about  equal  parts),  types  of  fringes  shown 
in  figure  63,  but  with  many  more  lines,  were  obtained.  This  is  a  combina- 
tion of  both  spindles  and  hyperbolae.  Probably  three  layers  of  liquid  are 
chiefly  in  question,  viz,  kerosene,  kerosene  +  CSa,  CS2,  and  the  three  stages 
of  form  and  the  sinuous  lines  correspond  to  them.  Fringes  were  sharp  only  if 
viewed  in  front  of  the  principal  focal  plane  of  the  telescope.  By  submerging 
convex  or  concave  lenses,  the  hyperbolic  parts  or  the  spindles  of  the  fringes 
could  often  be  removed.  Similar  results  were  obtained  with  mixture  of  CS2 
and  sweet  oil,  though  this  solution  is  more  homogeneous. 

50.  Apparatus. — To  obviate  the  tremor  of  apparatus  which  is  inevitable  in 
the  case  of  the  long-distance  interferometer,  the  parts  were  now  screwed 
down  at  short  distances  in  the  cast-iron  block  B,  figure  64.  Here  the  ranges 
MH,  HN  of  half-silvered  plate  H,  and  opaque  mirrors  M,  N,  did  not  exceed 


REVERSED   AND   NON-REVERSED   SPECTRA.  97 

14  cm.,  but  this  gives  ample  room  for  the  manipulation  of  the  trough  t  placed 
normally  in  the  beam  MH.  White  light  enters  by  way  of  the  collimator  SC 
at  any  convenient  angle  6  (as  this  does  not  enter  into  the  equations),  and 
0  =  6o°  was  used.  The  opaque  mirrors  M  (and  preferably  also  AT)  are  on 
micrometers  with  screws  normal  to  their  faces,  and  each  must  be  provided 
with  adjusting-screws  relatively  to  horizontal  and  vertical  axes.  An  elastic  fine 
adjustment  is  desirable.  The  block  contains  a  number  of  screw-sockets,  6, 
for  attaching  subsidiary  apparatus.  The  trough  t  should  preferably  be  at- 
tached to  an  independent  supporting  arm,  not  connected  with  B,  and  be 
revolvable  about  two  axes  normal  to  each  other.  In  such  a  case  the  position 
normal  to  the  beam  of  light  may  be  found  from  the  reverse  of  motion  of  the 
interference  rings,  while  the  trough  is  slowly  rotated  in  a  given  sense. 

The  telescope  T  (relatively  much  enlarged  in  the  diagram)  is  not  attached 
to  the  block.  It  is  to  be  used  both  as  a  simple  telescope  for  the  adjustment 
of  the  white  slit  images  to  horizontal  and  vertical  coincidence,  and  as  a  direct- 
vision  spectroscope.  The  most  convenient  attachment  for  this  purpose  is 
the  direct-vision  prism  grating  G  (film  grating)  just  in  front  of  the  objective 
of  T.  Two  perforated  thin  disks  of  brass  are  useful  for  this  purpose,  one  disk 
being  firmly  attached  (like  the  cap)  to  the  objective,  the  other  to  the  flat 
face  of  the  grating  with  the  prism  outward.  A  swivel  bolt  a,  between  the 
disks,  thus  allows  the  observer  to  throw  out  the  grating  and  use  the  telescope. 
A  stop  arrests  the  motion  of  the  grating  when  it  is  rotated  about  a,  back 
again,  for  viewing  the  spectrum.  This  plan  worked  very  well,  and  the  el- 
lipses obtained  were  magnificent.  It  was  almost  possible  to  control  the  microm- 
eter M  manually,  and  all  hurtful  quiver  is  absent.  The  fiducial  line  to  which 
centers  of  ellipses,  etc.,  are  to  be  returned  is  always  the  sodium  doublet  pres- 
ent in  sunlight  and  the  arc  and  artificially  supplied  by  an  interposed  burner 
in  case  of  the  Nernst  lamp.  The  telescopic  lens  need  not  be  more  than  2  cm. 
wide,  and  cross-hairs  are  not  needed.  For  measuring  dispersion  the  Fraun- 
hofer  lines  B,  C,  D,  E,  b,  F  were  used. 

51.  Equations. — The  useful  equations  for  present  purposes  are  given  in  a 
preceding  report,*  and  the  following  cases  only  need  be  repeated  here.  If  e 
is  the  thickness  of  glass  plate  of  index  of  refraction  n  for  the  wave-length  X, 
and  if  the  equation  M=^+£/X2,  where  A  and  B  are  constants,  be  taken  as 
sufficient, 
(i)  n-i=AN/e-2B/\* 

where  AAT  is  the  displacement  of  the  micrometer  at  the  opaque  mirror  M  or 
N  due  to  the  insertion  of  the  plate  normally  to  the  component  beam  in  ques- 
tion. To  determine  //,  B  must  be  known  at  least  approximately.  It  may  be 
measured  in  the  same  adjustment,  however,  if  two  Fraunhofer  lines  are  used 
fidutially.  Let  8N  be  the  displacement  of  micrometer  to  pass  the  center^of 

*  Carnegie  Inst.  Wash.  Pub.  No.  229,  1915.  §§  4<>,  4',  42- 


THE   INTERFEROMETRY  OF 


ellipses  from  wave-lengths  X  to  X'.    Then  if  e'  is  the  thickness  of  the  half- 
silvered  plate  H,  and  R  the  angle  of  refraction  within  it, 

(2)  dN  = 

If  <?=o, 

(3) 


which  may  be  called  the  corresponding  air  displacement.    Hence 

8N-dNa 
(4) 


B 


Here  8N  =  dNa=N-N'-(Na-N'a)  =N-Na-  (N'-N'a) 

so  that  the  difference  of  the  corresponding  positions  of  micrometer  for  a 
given  Fraunhofer  line,  with  and  without  the  plate,  are  to  be  found.  The 
method  is  quite  accurate,  as  will  be  seen  below.  More  than  two  constants 
A  and  B  may  be  taken,  if  desirable. 

To  return  to  equation  (i),  remembering  that  2.8/X2  is  small,  it  is  seen  that 
the  percentage  accuracy  of  /*—  i  and  AW  are  about  equal.  Now  AN,  for  a 
plate  5  to  6  mm.  thick  and  ordinary  glass,  is  about  0.3  cm.  This  may  be 
measured  within  io"4  cm.  or  3  parts  in  io4  of  AN  or  one  or  two  units  in  the 
fourth  place  of  /z,  the  index  of  refraction  of  the  glass.  A  much  more  serious 
consideration  is  the  consistent  measurement  of  the  thickness  of  plate  e,  which 
must  be  given  to  io~4  cm.  if  the  same  accuracy  is  wanted.  Naturally  this 
presupposes  optic  plate.  Hence  the  data  below  will  be  inaccurate  as  to  abso- 
lute values  from  this  cause.  The  plates  used  frequently  showed  increases  of 
thickness  of  several  io~3  cm.  within  a  decimeter  of  length.  Absolute  values 
are,  however,  without  interest  in  this  paper. 

To  show  that  less  than  io~4  cm.  is  guaranteed  on  the  micrometer  in  the 
placing  of  elliptic  centers  at  the  sodium  line,  the  pairs  of  results  given  in 
table  22,  made  at  different  times  and  with  entirely  independent  adjustments, 
may  be  cited.  The  screw-pitch  was  0.025  crn-  an<i  the  drum  divided  into 
50  parts  with  a  vernier  to  o.i  part. 

TABLE  22. 


Fraunhofer 
line. 

Pitch,  drum. 

Pitch,  drum. 

Difference. 

B 

C 
D 
E 
b 
F 

x8s     17.1 
85    23.3 
85    41-0 
xx  86     14.9 
86     19.2 
/86     36.6 

74    33-2 
74    39-3 
75      7-0 
75    30.9 
75    35-2 
76      2.6 

0.25000+0.01695  cm. 
+  .01700 
+  .01700 
+  .01700 
+  -01703 
+  .01703 

x  Ellipses  long  horizontally,     xx  Circles.    /  Ellipses  long  vertically. 
The  total  difference  is  less  than  io"4  cm.  and  probably  due  to  the  width 
of  the  Fraunhofer  lines  with  deficiency  of  light  at  the  ends  of  the  spectrum. 


REVERSED  AND   NON-REVERSED   SPECTRA. 


99 


Apart  from  measurement  of  the  thickness  et  therefore,  the  method  is  guaranteed 
for  fourth-place  work. 

52.  Observations.— At  short  distances  the  ellipses  are  always  rounder  and 
present  in  all  focal  planes  of  the  telescope.  Doublet  lens  compensators  of 
relatively  large  focal  power  are  available.  Using  the  CSz-sweet  oil  solution, 
the  submergence  of  convex  and  concave  lenses  in  it  made  but  very  little  dif- 
ference in  the  position  and  definition  of  the  ellipses.  There  must,  therefore, 
have  been  approximate  equality  of  the  conditions  of  refraction.  With  glass 
in  water  this  was  naturally  not  the  case. 

A  number  of  experiments  were  made  with  the  well-known  mercury-potassic 
iodide  solution  as  obtained  from  Eimer  &  Amend.  The  index  of  refraction 
in  the  concentrated  state  (floating  glass)  exceeds  1.7.  The  slight  straw- 
color  is  no  disadvantage. 

A  plane-parallel  trough  of  ordinary  plate-glass,  0.293  cm-  internal  width, 
was  now  constructed,  only  just  large  enough  to  receive  a  glass  plate  or  a  con- 
vex lens.  With  the  plate  or  lens  submerged  centrally,  therefore,  the  excess 
of  path-difference  over  the  empty  trough  would  be  nearly  constant.  The 
liquid  was  then  gradually  diluted  until  the  excess  of  path-difference  of  the 
liquid  over  the  glass  content  passed  through  zero  into  a  deficiency.  Con- 
siderable stirring  was  needed  to  insure  homogeneity  on  each  dilution.  The 
results  are  given  in  table  23  for  the  plate  and  in  table  24  for  the  lens.  So 
long  as  the  diluted  solution  was  effectively  more  refracting  than  the  glass, 
the  ellipses  were  nearly  circular  and  very  clear  both  for  the  submerged  plate 
and  lens,  as  well  as  for  the  liquid,  but  when  the  solution  began  to  effectively 
refract  less  than  the  glass,  the  ellipses  were  washed  and  could  not  be  obtained 
strongly.  On  submerging  the  lens  in  these  cases  it  was  necessary  to  so  adjust 
its  position  horizontally  and  vertically  that  the  white  images  (obviously  in 
different  focal  planes)  coincide  as  nearly  as  possible.  This  insures  a  sym- 
metric position  for  the  lens,  after  which  the  spectrum  is  to  be  examined  and 
the  ellipses  placed  by  moving  the  micrometer.  Unfortunately,  as  neither 
the  trough  nor  the  plate  was  optically  plane  parallel,  some  readjustment 
for  this  was  necessary  at  each  observation,  an  operation  which  introduces 
the  error  in  the  results  shown  in  tables  23  and  24,  so  far  as  absolute  values 
are  concerned,  which  has  been  alluded  to  above. 

TABLE  23.— Submergence  of  plate-glass  (thickness  0.284 
cm.)  in  mercury-potassic  iodide  solution.  Thickness  inside 
of  trough,  0.293  cm-  Sodium  line.  2B/\*=K.  Glass, 
5  =4.5X10-";  2S/X2  =  0.0262. 


Solution. 

Liquid 
AN 

Glass  in 
AN 

Liquid 
M-i+K 

Glass  in 
»-i+K 

6 

i 

cm. 
0.1640 
.1570 
.1494 

cm. 
0.1468 
.1466 
.1466 

0.6064 
.5824 
.5564 

0.5476 
.5469 
•5469 

9 

.1478 

.1468 

.5505 

•5475 

10 

ii 

.1430 
.1426 

.1466 
.1466 

•5345 
•5333 

•5470 
.5468 

100 


THE   INTERFEROMETRY  OF 


TABLE  24. — Submergence  of  glass  convex  lenses  in  mer- 
cury-potassic  iodide  solutions.  Thickness  of  the  trough 
inside,  0.293  cm.  Focal  power,  i  diopter.  Sodium  line. 
K  =  2B/\*.  Glass,  5=4.5X10-";  2S/X2  =  0.0262. 


Solution. 

Liquid 
AN 

Glass  in 
A# 

Liquid 

n-i+JC 

Glass  in 
n-i+K 

cm. 
0.2069 

cm. 

O.7525 

2 

3 
4 
5 

.1841 

.1692 

•1599 
.1412 
.1411 

x  0.1722 
.1742 
.1638 
•1565 
xx    .1432 

/  -1445 

.6746 

.6239 
•5921 

•5283 
.5280 

0.6340 
.6410 

•6055 
•5805 
•5352 
•5396 

x  Focal  power  2  diopter,     xx  Washed  ellipses.    /  Concave  lens. 

In  both  these  series  the  trough  was  not  fixed  with  adequate  rigidity,  so 
that  errors  crept   in   from   this  source.     Nevertheless,  if  the  data  are  con- 


•53     -Si     -55     -56      -5T     -58     -59     -60 


•63      -65 


structed  graphically  (/i—  i+K  for  solution  as  abscissa  and  for  submerged 
glass  as  ordinate),  the  results  (figs.  65  and  66)  show  a  very  definite  trend 
and  show  also  that  slight  interpolation  would  be  possible.  It  would  be 
hasty,  however,  to  infer  that  the  intersection  at  a  of  the  graph  for  submerged 
glass  with  the  line  at  45°  through  the  origin  is  the  index  of  refraction  of  the 
glass,  in  so  far  as  the  data  for  the  liquid  are  trustworthy.  The  method 
seems  to  be  deserving  of  notice;  but  before  discussing  the  matter  further 
it  will  be  necessary  to  determine  the  dispersion  involved  in  B. 

53.  Dispersion  constants. — As  shown  in  §51,  the  constant  Bis  found  by 
passing  the  center  of  ellipses  between  Fraunhofer  lines,  both  in  the  presence 
and  absence  of  the  plate  to  be  tested.  In  the  case  of  liquids  the  empty  and 


REVERSED  AND   NON-REVERSED   SPECTRA. 


101 


the  filled  trough  are  similarly  compared.  Data  of  this  kind  for  a  dilute  solu- 
tion of  mercury-potassic  iodide  and  two  kinds  of  glass  are  given  in  table  25. 
Na  is  the  micrometer  reading  at  M  for  the  half-silvered  plate  alone.  In  case 
of  the  solution,  it  is  to  include  the  glass  plate  of  the  trough.  N-Na=AN. 

TABLE  25. — Dispersion  constants. 


Fraunhofer 
line. 

XXio6 

Na 

(i)  Solution. 
e  =  0.293  cm. 

N-Na 

(2)  Trough  glass. 
6  =  0.562  cm. 

N-Na 

(3)  Glass  plate. 
e  =  0.434  cm. 

N-Na 

cm. 

B 
C 
D 

68.7 
65-63 
58-93 

0.0152 
.0173 

.0220 

0.1530 
•1554 
.1622 

0.3008 

.3023 
•3065 

0.2390 
.2403 

.2436 

E 

52.70 

.0284 

•1731 

•3121 

.2481 

b 
F- 

51-77 
48.61 

.0296 
.0342 

•1753 
•  1851 

•3130 
•3171 

.2488 

.2520 

The  computed  values  n-i+K=(N-Na)/e=AN/e  are  given  in  the  graph 
figure  67  (series  numbered),  from  which  the  characteristic  difference  in  the 
dispersion  of  the  solution  and  of  the  glass  is  apparent. 


48      §0 


If,  now,  the  value  of  B  is  computed  by  equation  (4),  between  successive 
Fraunhofer  lines  B—E,C—b,  D—F,the  results  come  out  as  in  table  26  and 
the  coefficient  B  should  be  correct  to  about  i  per  cent.  For  instance,  in  the 
case  of  the  solution  (N— Na)\—  (N— Na)\'  =  &N= 0.0201,  0.0200,  0.0229, 
respectively. 

TABLE  26. 


Fraunhofer 
lines. 

(i)  Solution. 
5Xiou 

(2)  Trough  glass. 
5Xio» 

(3)  Glass. 
BXio" 

B-E 
C-b 
D-F 

15-38 
16.02 
19.22 

4-50 
4-49 
4.62 

4.71 
4-65 
4-76 

102  THE  INTERFEROMETRY  OF 

It  thus  appears  that  in  case  of  the  solution  the  equation  n=A+B/te  is 
far  from  sufficient  up  to  the  Fraunhofer  F  line.  Nevertheless  even  here  the 
mean  of  the  first  two  results  may  be  regarded  as  holding  in  the  region  of  the 
D  line.  The  large  value  of  B  as  compared  with  the  glasses  is  particularly 
noteworthy,  and  from  this  a  reason  for  the  poor  ellipses  obtained  with  dilute 
solutions  is  suggested.  If  for  two  media  the  &N/e  are  identical,  as  at  a  and 
b  in  figure  67,  this  implies  merely  that 


Hence,  when  the  coefficients  B  differ  as  largely  as  is  the  case  for  the  solution 
and  the  glass,  the  indices  /*  are  far  from  equal.  In  the  above  case,  if  n—  //  = 
0.071,  the  solution  should  show  no  displacement  at  the  D  line  when  the  glass 
is  submerged.  But  this  difference  in  the  indices  of  refraction  of  the  glass  and 
the  solution  is  enormous,  and  if  the  submerged  body  is  a  lens,  the  correspond- 
ing images  in  the  telescope  will  be  thrown  quite  out  of  focus.  Thus  the  el- 
lipses are  necessarily  washed.  Conversely,  when  the  indices  of  lens  and  solution 
are  nearly  equal  at  the  sodium  line,  the  displacement  is  2(eB—eB')/\z,  and 
therefore  considerable  (0.035  cm-  above,  in  the  example  taken)  ;  but  the  ellipses 
are  now  sharp  and  strong.  Unfortunately,  therefore,  displacement  is  no 
criterion  for  equality  of  n,  and  mere  dependence  on  the  sharpness  of  fringes 
is  insufficiently  accurate.  The  only  resource  left  is  to  compute  B  and  B'  for 
two  spectrum  lines  and  adjust  the  solution  for  this  displacement.  Again, 
this  is  not  convenient,  particularly  when  but  a  single  spectrum  line  is  at  hand. 

54.  Further  observations.  —  A  somewhat  wider  trough  was  now  constructed 
of  the  same  plate-glass  as  above.  The  following  dimensions  were  found  by 
calipering:  Glass  wall  plates,  top  0.290  cm.,  bottom  0.283  cm.;  internal 
thickness  (liquid  plate),  top  0.564  cm.,  bottom  0.563  cm.  As  the  light  passed 
through  near  the  bottom  of  the  trough,  the  second  data  are  to  be  taken  in 
each  case.  The  observations  were  made  with  sunlight  and  at  each  of  the 
Fraunhofer  lines  B,  C,  D,  E,  b,  F.  In  case  of  a  hazy  sun  the  lines  B  and  F 
were  often  less  strong  than  desirable;  but  in  other  respects  the  work  through- 
out progressed  smoothly,  showing  magnificent  ellipses  beginning  at  the  B 
line  with  the  horizontal  axis  longer  and  ending  at  the  F  line  with  the  vertical 
axis  longer.  Circles  occur  earlier  as  the  refraction  is  greater.  N—Na  in 
table  27  is  the  coordinate  referring  to  the  difference  of  micrometer  reading  for 
the  presence  and  absence  of  the  plate  under  observation.  If  the  glass  walls 
of  the  trough  alone  are  to  be  taken,  the  Na  refers  to  the  half-silver  plate  alone. 
If  the  solution  is  in  question,  Na  refers  to  the  half-silver  and  the  trough  walls 
taken  conjointly  and  in  place.  The  trough  is,  of  course,  not  to  be  moved; 
but  some  adjustment  is  needed  when  the  liquid  is  introduced,  which  mars 
the  absolute  result.  In  series  5  a  glass  plate  0  =  0.293  cm-  thick  was  sub- 
merged in  the  solution  without  readjustment.  Hence  of  the  resulting  re- 
fraction 293/563=0.5024  belongs  to  the  glass  and  270/563  =  0.4796  to  the 
solution  surrounding  it.  Table  16  contains  ten  series  of  results  with  the 


REVERSED   AND   NON-REVERSED   SPECTRA. 


103 


TABLE  27. — Refraction  of  glass  and  of  mercury-potassic  iodide  solutions.  New  trough 
Glass  walls,  6  =  0.283  cm.  thick,  each;  solution,  £  =  0.563  cm.  thick;  X  =  2B/X*. 
Figure  68  is  an  exhibit  of  the  more  important  of  these  results. 


Series. 

Fraun- 
hofer 
line. 

XXio« 

M.XIO* 

Atf 

M-X+X 

&NXI& 

BXio" 

(i)  Glass  plate. 
6  =  0.566  

B 
C 
D 

E 
b 

cm. 
68.7 
65-63 
58.93 
52.70 

5I.77 

cm. 
8.1 
9-7 
14.4 

*20.8 
22.0 

cm. 
0.3247 
•3263 
•3306 
•3364 
•3375 

0-5737 
.5766 
.5841 

•5943 
.5963 

cm. 
BE,  117 
Cb,   in 
DF,  1  10 

4-65 
4-65 
4.78 

F 

48.61 

26.6 

.3416 

.6035 

(2)  Dilute  solu- 
tion 

B 
C 
D 
E 

Same 
as 

.... 

0.2855 
.2899 
•3031 

\2V7 

0.5071 
•5H9 
•5384 

57^O 

BE,  382 
Cb,  382 
DF,  435 

15-26 
16.01 
19.00 

b 

above 

.3281 

.5828 

9* 

F 

.3466 

.61  56 

(3)  Glass  plate. 
6  =  0.566  cm... 

B 

C 
D 

E 
b 
F 

Same 
as 
above 

.... 

0.3178 
•3194 
•3237 
•3295 
•3305 
•3346 

0.5616 
•5643 
•5719 
.5822 
•5839 
•5912 

BE,  116 
Cb,   in 
DF,  109 

4-63 
4-63 
4-74 

(4)  Stronger 
solution  

B 
C 
D 
E 

Same 
as 

.... 

0.3098 
•3151 
•3304 
.1548 

0.5503 

•5597 
•5869 
•6303 

BE,  450 
Cb,  446 
DF,  524 

17.99 

18-75 
22.89 

b 

above 

.-1598 

.6391 

F 

.3828 

.6800 

(5)  Glass  plate  in 
solution  4. 

B 
C 
D 
E 

Same 
as 

:::: 

0.3077 
.3110 
.3207 

-1-1=6 

•05465 
•5697 

S062 

BE,  279 
Cb,  276 
DF,  309 

1  1.  io 
1  1  -59 
13-52 

6  =  0.293  cm... 

b 

above 

-1-187 

6oi6 

F 

^517 

624.7 

(6,  7)  Glass  plate. 

B 
C 
D 
E 

Same 
as 

8.0 
9-7 
14-5 

•03193 
.3207 
•3249 

0.5641 
.5666 
•5740 

egAI 

BE,  114 
Cb,   107 
DF,  108 

4-53 
4.48 
4.69 

e    0.5      cm... 

b 

above 

cgsS 

F 

26  8 

(8)  Strong  solu- 

B 
C 
D 

Same 
as 

!... 

0.4474 
.4582 
.4916 

0.7947 
.8139 
.8732 

BE,  1000 
Cb,  1003 
DF,  1224 

39-94 
42-03 
53-47 

n  

above 

eege 

F 

.... 

.6140 

I.O9O6 

(9)  Strong  solu- 

B 
C 
D 

Same 
as 

0-4474 
.4584 
.4918 

0-7953 
•8l43 
.8736 
0727 

BE,   999 
Cb,  1004 
DF,  1216 

39-90 
42.06 
53-H 

above 

.... 

b 

•5589 

F 

(10)  Dilute  solu- 
tion.  .  . 

B 
C 
D 

E 

Same 
as     < 

.... 

0.3255 
.3309 
.3487 
•3764 

0.5782 
.5879 
•6194 
.6687 

BE,  509 
Cb,  5H 
DF,  601 

20.32 
21.56 
26.25 

b 
F 

above 

.3824 
.4088 

.6792 
.7261 

>  Circles. 


104 


THE   INTERFEROMETRY   OF 


same  trough  but  with  different  solutions,  dilute  and  nearly  concentrated. 
To  place  the  trough  normal  to  the  beam,  the  reflection  of  the  latter  from 
the  face  of  the  trough  was  made  to  coincide  with  the  spot  on  the  half-silver. 
This  is  inadequate  for  fine  work,  but  the  interferometer  method  of  retro- 
gressing fringes  is  not  applicable  unless  the  trough  is  separately  mounted. 
From  the  data  obtained  the  constant  B  was  computed  from  pairs  of  Fraun- 
hofer  lines,  B  and  E,  C  and  b,  D  and  F.  Apart  from  the  effect  of  the  thick- 
ness e,  it  should  be  correct  in  case  of  the  solutions  to  about  a  few  tenths  per 
cent.  In  case  of  the  concentrated  solutions,  however,  the  ellipses  already 
begin  to  move  sluggishly  and  are  much  smaller. 


70 


From  ju—  I+2.S/X2  and  B,  the  corresponding  indices  of  refraction  in  table  27 
are  easily  computed ;  but  these  data  are  of  little  value  here,  because  the  thick- 
ness e  of  the  ordinary  plate-glass  used  is  not  constant  to  the  degree  necessary. 
It  is,  however,  worth  while  to  exhibit  the  B  for  the  different  mercury-iodide  solu- 
tions in  terms  of  either  p  or,  what  is  equally  serviceable  and  more  convenient, 
in  terms  of  /z—  i+K;  for  this  quantity  is  directly  given  by  the  displacement 
measurement  bN/e.  If  we  regard  the  mean  B  for  the  B  to  E  lines,  and  C  to 
b  lines  as  applying  to  the  D  line,  the  coordinated  values  are: 

(2)  (4)  (8)  (9)  do) 

H  —  i+K 0.5384        0.5869         0.8732        0.8736        0.6194 

BXio11 15.63          18.37          40.99          40.98          20.94 

For  solutions  of  small  concentration  the  ratio  of  B  and  n—  i+K  changes 
but  slowly  (29  to  34)  as  shown  in  figure  69,  so  that  B  may  be  predicted  from 
the  latter,  always  remembering  that  our  equation  with  two  constants  (n=A 
+B/X2)  is  inadequate  at  the  outset. 


REVERSED  AND   NON-REVERSED   SPECTRA. 


105 


To  take  the  case  of  the  submerged  glass-plate  series  5,  if  e'  be  the  thick- 
ness of  plate  and  e"  of  solution,  so  that  *'+*"=*,  the  thickness  of  trough, 


Thus,  0.5204  of  the  data  of  series  (3)  added  to  0.4796  of  the  data  of  series 
(4)  should  reproduce  series  (5).    The  results  are  given  in  table  28. 

TABLE  28. 


B 

C 

D 

E 

b 

F 

3  1  Computed  
5    Found  

.  Difference.  .  .  . 

0.5561 
.5465 

0.5621 
•5525 

0-5791 
•5697 

0.6052 
.5962 

0.6104 
.6016 

0.6338 
.6247 

.0096 

.0096 

.0094 

.0090 

.0088 

.0091 

These  differences  are  nearly  constant  and  due  to  the  orientation  of  the 
glass  plate  with  which  its  effective  thickness  (assumed  0.293  cm-}  will  vary. 

55.  Conclusion.  —  It  appears,  therefore,  that  the  expectation  of  recognizing 
the  equality  of  refraction  of  a  submerged  solid  and  a  solution,  at  any  given 
wave-length,  from  the  fixity  of  the  fringes  in  the  presence  and  absence  of 
the  solid  has  not  been  fulfilled,  at  least  for  the  mercury-potassic  iodide  solu- 
tion. The  reason  is  found  in  the  enormous  difference  of  the  dispersions  of  the 
solution  and  ordinary  glass.  When  the  ellipses  are  not  displaced,  n~n'  = 
2(B'—B)/\2,  and  this  difference  may  even  approach  a  unit  in  the  first  decimal 
of  p..  The  troughs  in  which  such  experiments  are  to  be  made  must  be  optically 
plane  parallel,  as  otherwise  an  inadmissible  error  due  to  thickness  of  plates 
is  introduced.  With  such  a  trough,  however,  the  ease  and  accuracy  with 
which  the  dispersion  constants  may  be  found,  at  least  for  the  solution,  are 
noteworthy. 

When  the  solution  is  more  refracting  than  the  glass,  it  is  curious  that  the 
ellipses  are  not  seriously  distorted  or  vague,  even  when  the  symmetrically 
submerged  solid  is  lenticular.  Hence  the  equation  just  stated  is  available 
for  a  wide  variation  of  form.  Furthermore,  if  AAf  is  the  displacement  at 
the  micrometer  corresponding  to  the  presence  and  absence  of  glass  of  the 
thickness  e, 


But  as  /*',  B'  for  the  solution  are  known,  M  and  B  for  the  glass  may  both  be 
computed  from  observation  at  a  number  of  wave-lengths,  X,  provided  M  = 
A+B/\Z  for  glass,  which  is  sufficiently  nearly  so  to  the  fourth  place  of  decimals. 
Hence  if  &N/e+n'+2B'/\2=x  is  known, 


106 


INTERFEROMETRY  OF   SPECTRA. 


and  /i=#+2.B/X2.  Data  of  this  kind  are  given  in  table  29,  the  absolute  values 
depending  wholly  on  the  correctness  of  the  thickness  E  of  the  solution  and 
e  for  the  submerged  lens.  The  ellipses  were  excellent,  sharp,  and  strong. 
/t—  i+2.B'/X2  is  found  directly  and  AN  is  negative  as  the  solution  is  stronger. 

TABLE  29. — Refraction  of  lenses.    D  line.    Thickness  of  solution,  £=0.563 
cm.;K'  =  n'  —  i-f-2-B'A2;  2.B/X2  =  0.0260  for  glass.    n—l=K'—AN/e. 


Lens. 

e 

K' 

AWXio3 

M 

Remarks. 

I  diopter  
2  diopters.  .  .  . 
5  diopters  
10  diopters  

cm. 
0.138 

.200 

.248 
•447 

0.6140 
.6140 
.6140 
•6343 

cm. 
7.8 
10.7 
•18.7 

*20.5 

I-53I5 

1-5345 
1.5126 
1-5625 

Ellipses  strong. 
Ellipses  strong. 
Ellipses  vague. 
Ellipses  vague. 

*  Probably  not  adequately  centered. 

But  such  a  method  is  not  as  convenient  as  was  anticipated  at  the  outset, 
while  it  may  fall  short  of  the  needed  accuracy  if  rigorously  carried  out.  The 
only  escape  out  of  the  difficulty  would  be  the  use  of  a  liquid  of  about  the 
same  dispersion  constants  as  the  glass.  In  this  case  the  corrections  would 
be  minimized  and  the  experimental  work  simpler. 


CHAPTER  V. 


DISPLACEMENT  INTERFEROMETRY  IN  CONNECTION  WITH  U-TUBES. 
JAMIN'S  INTERFEROMETER. 

56.  Introduction. — A  variety  of  constants  in  physics  may  be  found  from 
the  relative  heights  of  two  communicating  columns  of  liquid.  This  is,  for 
instance,  the  case  in  the  classical  experiment  of  Dulong  and  Petit  on  the 
thermal  expansion  of  liquids.  Again,  if  one  of  the  tubes  is  subject  to  a  special 
force  acting  in  the  direction  of  its  axis,  this  force  in  its  bearing  on  the  liquid 
may  be  evaluated  from  the  resulting  difference  of  heads  of  the  columns. 
Thus  one  tube  may  be  surrounded  by  a  magnetizing  helix  and  the  effect  of 
the  axial  magnetic  field  on  the  liquid  in  question  (i.e.,  the  susceptibility) 
found  from  the  displacement  of  its  surface  by  the  presence  and  absence  of 
the  field,  etc.  It  seemed  to  me  worth  while,  therefore,  to  test  whether  it 
would  be  possible  to  measure  small  displacements  of  this  kind  by  passing  the 
two  component  beams  of  a  displacement  interferometer  axially  through  the 
two  columns  respectively,  and  to  measure  the  differential  effects  in  question 
in  terms  of  the  resulting  displacements  of  fringes. 

cfl 

70 


57.  Apparatus.  Michelson  interferometer.— The  interferometer  first  used 
was  of  the  same  form  as  that  described  above  (§  2,  fig.  3),  B,  figure  70,  being 
a  heavy  iron  block,  i  foot  in  diameter  and  1.5  inches  thick,  on  which  the 
mirrors  M,  N  (the  latter  and  preferably  both  on  micrometers)  are  securely 
mounted  with  the  usual  direct  rough  and  elastic  fine  adjustment  for  hori- 
zontal and  vertical  axes.  A  beam  of  parallel  white  rays  L  arrives  from  a 
collimator  (not  shown)  and  impinges  on  the  half-silver  plate  //,  to  be  reflected 
and  transmitted  at  a  convenient  angle  6  (about  60°),  thus  furnishing  the  two 
component  beams  which  are  to  traverse  the  limbs  of  the  U-tube. 

107 


108  THE   INTERFEROMETRY   OF 

The  vertical  columns  of  this  tube  are  shown  at  C  and  C'  (with  accessory 
mirrors  removed),  and  they  are  joined  to  the  capillary  tube  p  near  the  bottom 
of  C  and  C'.  Details  will  be  given  in  connection  with  figures  71  and  72. 

The  ray  HM  strikes  a  mirror  symmetrically  at  45°  to  the  vertical  below  C, 
is  thence  reflected  upward  along  the  axis  a,  striking  another  mirror  above, 
also  symmetrically  at  45°  and  parallel  to  the  former,  whence  it  is  reflected 
to  the  opaque  mirror  M.  The  latter  reflects  the  ray  normally  back,  so  that 
it  retraces  its  path  as  far  as  H,  by  which  plate  it  is  now  transmitted  to  be 
observed  by  the  telescope  at  T.  Similarly  the  transmitted  component  ray 
HN  is  guided  by  suitable  reflectors  at  45°,  so  as  to  take  the  path  Ha'Na'HT, 
thus  passing  axially  (a')  through  the  tube  C'. 

It  is  necessary  that  the  U-tube  CpC'  be  mounted  independently  of  the 
block  B  on  suitable  bracket  or  arm  attached  to  the  pier.  Otherwise  any 
manipulation  at  N  will  disturb  the  surfaces  of  water  in  C  and  C'.  Ordinary 
clamps  admit  of  raising  or  lowering  or  rotating  CC'  satisfactorily,  always 
providing  that  it  shall  not  touch  B.  The  telescope  at  T  is  also  mounted  apart 
from  B  on  the  table  below.  The  direct-vision  prism  grating  g  is  placed  im- 
mediately in  front  of  the  objective  and  swiveled  as  described  in  figure  64, 
Chapter  IV,  so  that  either  the  white  slit  images  or  their  spectra  may  be  seen 
in  the  field  of  view,  according  as  g  is  rotated  aside  or  is  in  place. 

In  figure  71  a  front  sectional  elevation  of  one  of  the  shanks  of  the  U-tube 
is  given  with  all  appurtenances,  and  a  similar  sectional  elevation  at  right 
angles  to  the  former  is  added  in  figure  72  for  the  top  of  the  tube.  In  figure  7 1 
the  mirrors  m'  and  m  are  on  horizontal  axes  and  the  component  ray  coming 
from  behind  the  diagram  strikes  m'  below,  is  reflected  axially  upward  through 
CC,  impinging  on  the  mirror  m  (also  on  a  horizontal  axis) ,  whence  it  is  reflected 
horizontally  toward  the  front  of  the  diagram.  The  ray  a  and  mirror  m  are 
given  more  clearly  in  figure  72.  The  lateral  capillary  tube  appears  at  p  and 
the  tube  C  is  closed  below  with  a  plate  of  glass  e,  cemented  in  place. 

To  mount  the  mirrors  m,  m',  snugly  fitting  rings  r  and  r'  encircle  the  tube 
C  near  its  top  and  bottom  and  can  be  fixed  by  the  set-screws  5  and  s'.  In 
virtue  of  these  rings,  the  mirrors  m,  m'  may  be  rotated  at  pleasure  around 
the  vertical  axis  a  of  CC.  The  horizontal  axis  of  the  mirrors  m,  m'  rotates 
at  pleasure  in  the  vertical  arms  A,  A'  of  square  brass  tube.  A,  A'  m  turn  may 
be  slightly  swiveled  about  the  horizontal  axis  b,  b',  in  a  rigid  lateral  projection 
of  the  rings  r,  r' .  Thus  m,  m'  are  capable  of  rotation  around  three  axes  normal 
to  each  other  and  adequately  clamped  in  any  position. 

The  component  ray  HN  may  be  adjusted  to  the  center  of  the  lower  mirror 
m'  by  placing  the  collimator  L  and  then  guided  axially  by  m',  m,  N  as  described, 
each  being  adjustable.  The  component  ray  HM  may  be  similarly  adjusted 
to  the  center  of  the  lower  mirror  m'  (at  45°)  by  slightly  rotating  the  half- 
silver  plate  H  (on  horizontal  and  vertical  axes)  and  then  guided  axially  by 
mr,  m,  M.  As  a  whole  the  adjustment  is  difficult,  though  it  need  not  be  much 
refined.  Clear  white  slit  images  in  the  telescope  T  are  an  adequate  criterion. 

In  the  absence  of  a  liquid  in  CC,  figure  71,  the  fringes  are  easily  found  after 


REVERSED  AND   NON-REVERSED   SPECTRA.  109 

careful  preliminary  measurement,  and  they  are  strong  and  satisfactory. 
When  this  adjustment  is  given,  the  presence  of  liquid  in  CC,  if  the  two  columns 
are  of  nearly  equal  length,  does  not  much  modify  the  adjustment.  In  fact, 
the  fringes  were  found  much  more  easily  than  I  anticipated,  and  in  quiet 
surroundings  they  are  strong  and  fine.  It  is  necessary,  however,  that  the 
tube  CC  should  be  of  sufficient  width  to  avoid  all  curvature  due  to  capillarity, 
at  least  in  the  axis.  Tubes  2  cm.  in  diameter  and  10  cm.  long  of  thin  brass 
were  first  tried,  but  proved  to  be  too  narrow.  No  sharp  slit  images  could  be 
obtained  with  reasonable  care  as  to  setting  the  mirrors.  Thereafter  tubes 
4  cm.  in  diameter  were  used,  but  even  these  are  somewhat  too  narrow.  Slit 
images,  however,  were  sharp  and  parallel  and  could  be  easily  brought  to 
coincide. 

With  the  wide  tubes,  however,  the  mobility  of  the  liquid  in  CC  increases 
enormously,  so  that  only  under  exceptionally  quiet  conditions  could  the 
fringes  be  seen,  and  never  quite  without  quiver.  The  wind  beating  on  the 
house,  for  instance,  threw  them  nearly  out  of  view,  so  that  only  a  suggestion 
of  their  presence  remained.  In  spite  of  the  very  promising  beginnings,  there- 
fore, it  became  a  serious  question  whether,  with  the  apparatus  as  here  devised, 
the  purposes  of  the  research  could  be  reached  in  this  laboratory. 

Finally,  the  flickering  of  the  arc  lamp  may  be  a  grave  inconvenience;  for 
if  the  columns  C,  C'  as  usual  are  virtually  prisms,  the  coincidence  of  spectra 
will  for  this  and  other  reasons  be  destroyed  by  the  displacement  of  the  arc. 

58.  Equations.—  Some  estimate  of  the  increments  to  be  anticipated  may 
be  given  here,  and  expressed  in  terms  of  the  Dulong-Petit  experiment.  If  a 
is  the  mean  coefficient  of  expansion  of  water  at  the  temperature  in  question 
and  A//  the  increment  of  the  head  H  corresponding  to  the  temperature 
difference  A*  between  the  columns, 

(i)  &H=aHM 

Again,  if  A/V  corresponding  to  AH  is  the  displacement  of  centers  of  ellipses 
at  the  wave-length  X,  and  n  the  index  of  refraction  of  water,  so  that  M  =  A  +B/\*, 
nearly, 

AAT 


Hence  A*  may  be  computed  as 

AAT 


Since  the  value  of  AAT  is  within  iQ-4  cm.  and  H=  10  cm.  in  the  above  appa- 
ratus, we  may  further  write  at  mean  temperatures  (25°) 
a  =  2.5Xio^         M=i.333          £  =  io-llX3.i          2£/X*=  0.018  at  the£>  line. 
Thus  Ai-i-f-2B/X2=o.35i   and  A*=  10^/0.351  X2.sXio-4Xio  =  o.ii4°. 
other  words,  in  case  of  tubes  10  cm.  long,  the  effect  of  a  difference  of  tempera- 


110  THE   INTERFEROMETRY  OF 

ture  of  about  o.i  degree  between  the  tubes  should  be  easily  observable  by 
mere  displacement,  whereas  a  difference  of  less  than  0.03  would  be  equivalent 
to  the  passage  of  one  interference  ring. 

Again,  from  equation  (2),  if  AN=  icr4  cm.,  then  AH  =  io~*/o.35i  =  lo^Xa.S 
cm.,  or  about  gX  icr6  cm.  per  vanishing  interference  ring  are  the  displacements 
to  be  anticipated.  These  are  equivalent  to  pressures  of  about  0.3  and  o.i 
dyne  per  square  centimeter. 

59.  Observations.  —  A  large  number  of  observations  were  made  with  the 
apparatus  described,  but  as  under  present  surroundings  the  fringes  always 
quivered  violently,  no  quantitative  results  of  value  were  obtained.  Naturally 
such  experiments  imperatively  demand  a  laboratory  remote  from  traffic,  since 
the  undulation  of  mobile  liquid  surfaces  is  introduced  in  addition  to  the  tremors 
of  solid  appurtenances. 

An  attempt  was  made  to  register  the  pressure  near  an  electrically  charged 
point,  but  no  results  could  be  obtained.  Again,  though  the  attraction  of  an 
electrically  charged  surface  for  the  free  surface  of  water  in  either  tube  was 
recognized,  on  using  adequately  high  potentials  to  measure  the  forces  the 
surface  became  troubled  and  the  fringes  vanished.  In  this  case,  if  p  is  the 
pressure,  V  the  difference  of  potential  in  volts,  and  d  the  distance  apart  of 
surfaces  in  centimeters,  /=4.4Xio~7X(V/c02  dynes/cm.2  if  7=0.3  can  just  be 
determined  by  the  displacement  method,  and  if  V  =  80  volts  (roughly)  is  the 
smallest  potential  difference  discernible  for  quiet  fringes.  Finally,  Dulong 
and  Petit  's  experiment  gave  very  definite  results  even  for  small  ranges  of 
temperature,  subject  to  the  conditions  stated. 

By  surrounding  the  top  of  the  tube  C,  figure  71,  with  a  close-fitting  helix, 
the  upper  face  of  which  reached  just  below  r,  while  the  surface  of  liquid  within, 
w,  lay  at  its  center,  an  attempt  was  made  to  detect  the  susceptibility  k  of 
water,  etc.  If  the  field  of  the  coil  be  written  roughly  H=o.4irin/l,  where  i 
is  the  current  in  amperes,  n  the  number  of  turns,  and  /  the  length  of  the  helix, 
we  may  write 


Since  p  =  i,  the  increment  of  head,  h,  becomes 


Hence  if  roughly  k  =  io-*,  g=io8,  i=i  am.,  «//=3S  as  in  the  helix  used, 
A/*  =  442X  io-9=  i. 


Thus  if  i=io  amp&res,  A/t  =  2Xio~4  cm.,  nearly,  and  easily  determinable  by 
the  displacement  of  ellipses,  or  from  interference  rings.  The  experiment  was 
tried,  but  the  quiver  of  rings  was  such  as  to  admit  of  no  decision.  In  case  of 
the  magnetic  solutions,  k  is  of  course  much  larger;  but  under  the  circumstances 
it  did  not  seem  worth  while  to  attempt  further  work.  This  will  be  done  with 
other  apparatus  in  the  course  of  this  paper. 


REVERSED  AND  NON-REVERSED  SPECTRA. 


Ill 


60.  Jamin's  interferometer.— The  ease  with  which  the  Michelson  inter- 
ferometer may  be  adjusted  and  its  remarkable  adaptability  have  led  to  its 
general  preference  over  the  older  form  of  Jamin.  Nevertheless,  the  latter 
furnishes  two  parallel  rays  which  for  such  purposes  as  the  present  are  desir- 
able. Hence  if  the  four  faces  of  the  interferometer  be  separated  in  the  manner 
suggested  by  Mach  (shown  in  fig.  73),  a  very  available  form  of  interferom- 


eter is  obtained.  Here  M  and  N  are  half-silvered  plates,  M'  and  N'  the 
opaque  mirrors.  The  white  light  L  impinging  from  a  collimator  thus  fur- 
nishes the  two  component  beams  ac  and  bd,  which  are  observed  with  the  tele- 
scope at  T,  after  passing  the  direct-vision  prism  grating  g.  If  either  mirror 
M'  or  N'  is  displaced  a  distance  e,  moving  parallel  to  itself,  the  path-differ- 
ence 2e  cos  6  is  introduced  with  the  corresponding  shift  of  ellipses.  The 
U-tubes  C,  C',  with  their  helices  H,  H',  and  connecting  pipe  p  are  now  con- 
veniently installed  as  shown.  But  the  trouble  with  the  arrangement  is  the 
difficulty  of  adjusting  the  Jour  surfaces.  Not  only  are  the  centers  of  ellipses 
liable  to  be  remote  from  the  center  of  the  field,  but  it  is  often  hard,  without 
special  equipment,  to  even  find  the  fringes. 

If,  however,  the  device  which  I  suggested  in  the  preceding  report  is  adopted — 
i.e.,  if  (fig.  74)  the  half-silvered  plates  M,  N  are  at  the  ends  of  a  single 
strip  of  plate-glass,  so  that  rays  terminating  in  M,  M'  N  N'  after  adjust- 
ment necessarily  make  a  rhombus-like  figure  symmetrical  to  MN — the  fringes 
are  found  at  once;  for  they  appear  when  the  white  slit  images  in  T  coincide 
horizontally  and  vertically  and  the  rays  bd  and  cd  intersect  in  the  common 
point  d.  Hence  the  mirrors  M',  N'  should  be  on  carriages  D,  F,  adapted  to 
move  on  parallel  slides  5,  5'.  M,  N  may  also  be  put  on  a  carriage  E,  though 
this  is  not  necessary.  S,  5'  need  not  be  parallel  to  ac  or  bd.  If  the  mirrors 
M'  and  Nf  are  wide,  considerable  latitude  of  adjustment  is  thus  obtained. 

If  MN  is  half-silvered  on  the  same  side  (*.*.,  toward  N')  a  compensator  is 
needed  in  ac  or  cd  if  path-difference  is  to  be  annulled  (symmetry).  If,  how- 
ever, M  is  half-silvered  on  the  N'  side  and  N  on  the  M'  side,  no  compensator 
is  required.  In  the  latter  case,  however,  if  ordinary  plate-glass  is  taken,  M 
and  N  are  not  quite  parallel  and  the  ellipses  will  be  eccentric.  This,  however, 
is  not  necessarily  a  disadvantage,  unless  the  strip  MN  is  excessively  wedge- 
shaped. 


112  THE   INTERFEROMETRY   OF 

The  ellipses  obtained  are  usually  long  vertically  —  i.e.,  quite  eccentric  — 
so  that  the  fringes  soon  become  straight  and  the  rotation  is  extremely  rapid 
whenever  the  center  of  ellipses  is  out  of  the  field.  It  is  therefore  possible  to 
adjust  relative  to  horizontal  fringes  (parallel  to  the  shadow  of  wire  across 
slit),  as  these  incline  very  obviously  for  a  displacement  of  less  than  io~*  cm. 
and  rapidly  become  vertical.  For  this  reason  it  makes  little  difference  in 
practice  whether  the  half  -silvers  are  on  the 
same  or  on  opposite  sides,  or  whether  obser- 
vation be  made  at  T  (cd  prolonged)  or  at  T' 
(bd  prolonged).  Moreover,  the  plate  MN 
may  be  conveniently  constructed,  as  in  figure 
75,  of  two  mirrors  m,  n,  attached  to  the  clear 
strip  of  plate-glass  g  by  aid  of  strong  steel  76 

clips  at  c,  c'.  With  the  half-silvers  5,  5',  on  the  same  side,  the  wedge-angle 
of  the  glass  is  excluded.  For  shorter  diagonals,  the  plan  of  figure  76,  with 
the  silver  surfaces  s,  s'  held  together  by  clips  at  c,  is  preferable. 

If  the  mirror  M',  figure  74,  is  displaced  a  distance  e,  where  a  glass-plate 
compensator  of  thickness  E  and  refraction  constants  n  and  B  is  introduced 
normally  either  into  ab  or  bd,  the  equation  is  easily  seen  to  be,  at  wave- 
length X, 


where  B  is  the  angle  of  reflection  at  M.  Using  the  plate  £  =  0.434  cm.  treated 
above,  the  first  member  is  0.2428  cm.  Values  of  e  of  0.2420,  0.2409,  0.2427 
were  roughly  obtained.  Hence  the  mean  value  of  6  should  be  about  60°, 
as  it  actually  was. 

The  occurrence  of  this  angle  and  the  shift  of  the  beam  bd  along  the  mirrors 
M'  and  N  are  the  main  objections  to  the  method  of  figure  74,  for  the  rhom- 
bus is  not  necessarily  perfect.  If  the  ends  of  the  plate  MN  are  silvered  on 
the  same  side,  the  compensator  must  have  double  the  plate-thickness  to 
annul  path-difference.  Finally,  the  half-silvering  does  not,  for  large  6,  suffi- 
ciently exclude  the  reflection  of  bd  from  the  naked  face  of  the  plate,  so  that 
the  fringes  are  never  quite  black.  These  difficulties  may  be  met  by  making 
MN,  figure  74,  the  short  diagonal  of  the  rhombus  and  using  the  strip,  figure 
76.  In  such  a  case  6  at  M'  is  small,  and  in  view  of  the  nearly  normal  reflec- 
tion at  M  and  N  relatively  little  reflection  comes  from  naked  glass,  sliding 
is  largely  avoided,  and  no  compensator  is  necessary.  In  this  case  the  fringes 
for  no  path-difference  are  actually  strong  black  horizontal  lines  on  a  colored 
ground  and  far  enough  apart  that  o.i  fringe  could  easily  be  estimated.  A 
test  experiment  with  the  above  plate  showed  €  =  0.1244  cm.,  corresponding 
to  the  small  angle  6,  a  little  over  12°. 

When  the  U-tube  CC',  figures  71  and  74,  is  introduced,  the  strip  MN  will 
have  to  be  at  a  considerable  angle  (about  45°)  to  the  horizontal,  so  as  to 
raise  the  N  end  about  15  cm.  above  the  M  end,  corresponding  to  the  height 
of  m  above  m'  in  figure  71.  The  new  condition,  however,  in  no  way  changes 


REVERSED   AND   NON-REVERSED   SPECTRA.  113 

the  general  procedure.  In  case  of  figure  74,  the  mirror  Nf  must  be  high  and 
M'  low.  This  is  usually  less  convenient  than  the  case  when  both  mirrors 
are  high  (C  placed  at  G)  or  where  both  mirrors  are  low  (Cr  placed  at  G'). 
In  the  former  case,  again,  the  rays  have  considerably  diverged  in  a  vertical 
plane  and  the  fringes  are  less  marked.  If  C'  is  at  G'  the  whole  of  each  com- 
ponent beam  may  be  caught  and  passed  through  the  respective  shanks  of 
the  U-tube.  The  fringes  are  strong,  easily  found,  and  large,  so  that  the  cen- 
ter of  ellipses  is  not  far  outside  of  the  field  of  the  telescope.  It  is  obvious 
that  to  facilitate  adjustment  the  mirrors  m  and  m',  figure  71,  must  be  nearly 
parallel.  They  are  made  so  by  the  aid  of  a  broad  beam  of  sunlight  and  then 
clamped  firmly  in  position  at  about  45°  to  the  respective  axis  of  the  U-tube. 

Finally,  if  the  connecting-tube  p  is  nearly  horizontal  when  in  place,  the 
fringes  are  usually  found  at  about  the  same  position  of  the  micrometer  (at 
M')  after  the  liquid  is  introduced  into  the  U-tube.  Here  it  is  advantageous 
and  usually  permissible  to  make  a  part  of  the  connection  p  of  flexible  rubber 
tubing.  But  unless  the  free  surface  w,  figure  71,  is  very  nearly  parallel  to 
the  plate  e,  the  center  of  ellipses  is  liable  to  be  far  outside  of  the  field  of  the 
telescope  and  the  fringes  correspondingly  small.  The  difficulty  of  adjust- 
ment for  large  fringes  is  now  considerable,  because  of  the  two  mobile  liquid 
surfaces  in  the  U-tubes.  For  this  reason  I  did  not  attempt  to  make  measure- 
ments, although  the  fringes  themselves  were  surprisingly  steady  and  strong 
and  would  have  been  quite  available,  apart  from  laboratory  tremors.  Slight 
changes  in  density,  due  to  solution  or  temperature  changes  in  one  shank  of 
the  U-tube,  were  well  recorded,  after  stirring,  with  curious  effects  of  surface- 
tension  and  viscosity. 

The  fringes  being  very  clear,  a  number  of  other  promiscuous  experiments 
were  tried.  Thus,  a  tube  with  plate-glass  ends  and  filled  with  water  was  made 
the  core  of  a  powerful  magnetic  helix.  The  tube,  26  cm.  long,  was  placed 
in  one  of  the  component  beams  and  compensated  by  a  column  of  glass  in 
the  other.  Good  fringes  were  easily  found;  but  not  the  slightest  displacement 
could  be  detected  by  alternations  of  presence  and  absence  of  the  magnetic 
field.  The  water  was  now  replaced  by  a  solution  of  nickel  sulphate.  Fringes 
were  again  easily  found  and  strong  in  the  green,  but  the  effect  of  the  magnetic 
field  was  quite  as  inappreciable  as  before.  Magnetic  fields  were  thus  totally 
ineffective. 

In  a  set  of  experiments  of  a  different  kind  the  attempt  was  made  to  observe 
the  gradual  deposition  of  silver  on  plate-glass.  Bottger's  solutions  were 
poured  into  a  plane-parallel  clean  glass  trough  normal  to  one  of  the  component 
beams  (of,  fig.  74)  and  compensated  by  a  plate  of  glass  in  the  other  (W). 
Large  fringes  were  produced  by  setting  the  micrometer  and  observed  during 
the  formation  of  the  two  silver  films  on  the  opposite  faces  of  the  trough,  until 
they  became  quite  opaque.  It  was  astonishing  to  find  that  fringes  were  still 
faintly  visible  long  after  a  highly  reflecting  mirror  had  been  deposited.  But 
no  displacement  larger  than  a  fraction  of  a  fringe  could  be  detected,  showing 
the  extraordinarv  thinness  of  the  silver  film  even  when  practically  opaque. 


114 


THE   INTERFEROMETRY   OF 


Similarly,  the  silver  deposit  on  plate-glass  was  removed  in  parallel  strips,  so 
that  the  film  had  the  appearance  of  a  grid.  The  results  when  this  plate  was 
placed  normally  in  one  of  the  component  beams  were  the  same. 

61.  Vertical  displacement  of  ellipses. — If  the  fringes  are  too  small  when 
horizontally  centered  by  the  micrometer,  the  center  of  ellipses  may  be  brought 
into  the  middle  of  the  field  of  the  telescope  by  sliding  one  component  beam 
vertically  over  the  other  without  appreciably  changing  the  direction  of  the 
rays.  In  other  words,  one  illuminated  spot  at  d,  figure  74,  is  to  move  vertically 
relative  to  the  other  by  a  small  amount.  This  may  be  done  by  placing  a 
thick  plate-glass  compensator,  such  as  is  shown  in  figure  77,  in  each  of  the 
component  beams  abd  and  acd  and  suitably  rotating  one  plate  relative  to  the 
other,  each  on  a  horizontal  axis.  Very  little  rotation  is  required.  In  the  same 
way  elliptical  fringes  may  be  changed  to  nearly  linear  horizontal  fringes 
when  desirable.  If  the  fringes  are  to  be  sharp  the  slit  must  be  very  fine.  When 
sunlight  is  used  with  a  slit  not  too  fine,  each  of  the  coincident  sodium  lines 
(AA)  frequently  shows  a  sharply  defined  helical  or  rope-like  structure,  the 
dark  parts  in  step  with  the  fringes  of  the  spectrum.  It  looks  like  an  optical 
illusion  of  slanting  lines  or  a  shadow  interference  of  two  grids  (fringes  and 
sodium  lines  respectively);  but  later  experiments  showed  it  to  be  an  inde- 
pendent phenomenon.  (Cf.  §  63  et  seq.,  68,  70.) 


a 


V  Hi  ( 


77  78 

The  first  result  is  particularly  interesting,  inasmuch  as  it  is  thus  possible 
to  displace  the  centers  of  ellipses  not  only  horizontally  as  usual  relative  to 
the  fixed  sodium  lines  in  the  spectrum,  but  also  vertically  relatively  to  the 
fixed  horizontal  shadow  in  the  spectrum  due  to  the  fine  wire  across  the  slit. 
The  following  experiment  was  made  to  coordinate  the  vertical  displacement 
of  the  component  rays  and  centers  of  elliptic  fringes:  A  glass  plate  d  =  0.705 
cm.  thick  was  placed  nearly  normally  in  the  beam  ac,  figure  74,  and  provided 
with  a  horizontal  axis  and  graduated  arc.  The  amount  (*)  of  rotation  of  the 
plate,  corresponding  to  the  vertical  displacement  of  one  central  fringe  in  the 
telescope  (i.e.,  passage  of  fringe  a  into  b,  into  c,  in  the  duplicate  spectrum  5, 
fig-  78),  was  then  found  to  be,  if  *  is  the  angle  of  incidence, 

i  h 

No  fringes 3.5°  0.0149  cm- 

One  fringe 5.0°  .0214 

Two  fringes 6.5*  .0281 

where  h  is  the  corresponding  vertical  displacement  of  the  rays  ac,  figure  74, 
and  computed  from  (p  index,  r  angle  of  refraction) 
h  =  d(sin  i  —  cos  i  tan  r) 


REVERSED  AND   NON-REVERSED   SPECTRA.  115 

Thus  the  vertical  displacement  of  rays  corresponding  to  the  vertical  semi- 
axes  of  the  central  ellipse  or  one  fringe  is  between  0.0065  and  0.0067  cm.— 
*.*.,  on  the  average  below  7  X  io'3  cm.  Hence  h=NXo.oo'j  for  N  such  central 
fringes.  It  was  difficult  to  get  a  closer  result,  owing  to  quiver. 

The  interesting  question  is  now  suggested,  in  how  far  such  an  arrangement 
would  fall  short  of  being  able  to  exhibit  the  drag  of  the  ether  in  a  rapidly 
rotating  body,  should  such  drag  occur.  In  figure  73,  let  aft  be  a  cylinder  of 
glass  with  plane-parallel  ends,  capable  of  rotating  on  the  axle  75.  If  /  is  the 
length  of  the  cylinder,  /x  its  index  of  refraction,  and  r  the  distance  of  either 
component  ray  (ac,  bd)  from  the  axis  yd,  n  the  number  of  turns  per  second, 
and  V  the  velocity  of  light,  we  may  write,  using  the  above  excessive  estimate, 
N  being  the  number  of  fringes  displaced, 


=  27rwr/M/F 
since  ac  rises  while  bd  falls.    If 
n=2oo,  r=iocm.,  /=ioo  cm., 


3.5  x  10-3x3  x  io'° 

It  would  thus  be  necessary  to  estimate  about  one-sixtieth  of  a  fringe,  which 
is  just  beyond  the  limit  of  certainty,  even  if  nr  can  be  increased  and  /  multi- 
plied by  reflection.  The  device  suggested  is  nevertheless  of  interest  and 
deserves  further  consideration.  It  will  appear  much  more  promising  in 
connection  with  the  achromatic  fringes  described  below. 

62.  Displacement  interferometer.  Jamin  type.  —  These  considerations 
induced  me  to  devote  further  study  to  the  Jamin  type  of  interferometer 
(fig.  73).  The  mirrors  M,  N'  were  put  on  one  pair  of  long  slides  (1.5  meters 
long)  parallel  to  ac  and  the  mirrors  M',  N  on  similar  slides  parallel  to  the 
former.  In  this  way  any  distance  ac  or  bd  was  available.  The  beams  were 
about  1  6  cm.  apart,  corresponding  to  a  normal  distance  between  the  end 
mirrors  (NN',  MM')  of  about  12  cm.  But  these  distances  could  also  be 
increased  from  nearly  zero  (M  and  M'  nearly  contiguous)  to  about  20  cm. 
in  view  of  the  width  of  mirrors  used.  The  angles  at  a,  b,  c,  d  were  each  about 
45°,  so  that  a  rectangle  of  rays  is  in  question.  (See  figure  88  or  93  below.) 

The  adjustment  proved  eventually  to  be  greatly  facilitated  by  using  a 
horizontal  beam  of  sunlight  with  weak  condenser-lens  and  collimator.  A  thin 
wire  is  to  be  drawn  across  the  dit.  M  and  M'  are  first  set  for  parallelism  in 
the  absence  of  N  and  N',  by  adjusting  the  images  of  the  slit  at  the  same  level 
(horizontal)  on  a  distant  wall.  The  images  or  shadows  of  the  wire  specified 
on  the  wall  are  to  be  equally  far  apart,  with  the  beams  ac  and  bd  at  the  mirror. 
The  mirrors  N  and  N'  are  next  put  in  place  with  the  distances  acd  and  abd 
about  equal.  The  two  images  seen  in  the  telescope  at  T  (g  removed)  are  then 
made  to  coincide  both  horizontally  and  vertically  by  adjusting  N  and  N  , 


116 


THE   INTERFEROMETRY  OF 


and  these  are  then  slid  by  a  small  amount  on  their  slides  (direction  ac)  until 
the  rays  are  coincident  at  d  to  the  eye  (light  strips  on  the  mirror  coincide). 

If,  now,  the  grating  g  is  inserted,  very  fine  oblique  fringes  will  usually  be 
seen.  These  may  be  enlarged  to  a  maximum  by  moving  the  micrometer 
controlling  the  displacement  M'  normal  to  itself.  Somewhat  coarser  horizontal 
lines  are  thus  obtained. 

Finally,  the  distant  centers  of  the  ellipses  are  brought  into  the  center  of 
the  telescope  by  aid  of  the  thick  glass  compensator,  like  figure  77  (the  equiva- 
lent air-path  of  the  other  ray  being  correspondingly  lengthened)  by  rotating 
the  glass  plate  on  a  horizontal  axis.*  It  is  desirable  to  have  an  excess  of  glass- 
path  in  one  beam,  as  otherwise  the  ellipses  are  so  large  as  to  be  unwieldy. 

The  ellipses  so  obtained  with  common  plate-glass  and  a  film  grating  at  g 
were  magnificent.  A  rough  test  of  the  displacement  interferometer  was  made 
by  using  the  above  plate-glass  of  thickness  £=0.434  cm.,  where  z=E(jji—  i) 
-f-2.B/X2  =  o.2428  cm.  In  two  experiments  agreeing  to  within  icr4  cm.,  20  = 
0.3448  cm.  were  the  displacements  obtained.  Assuming  that  0  =  45°,  26  cos  8 
=0.2438  cm.  This  agrees  with  z  as  nearly  as  may  be  expected,  unless  6  is 
specifically  measured. 

Experiments  were  now  made  (as  above)  with  thick  plate-glass  compen- 
sators inserted  in  one  component  ray  (bd)  only,  to  determine  the  rotation  of 
compensator  (i°)  necessary  to  raise  the  center  of  ellipses  in  steps  of  half  the 
diameter  of  the  first  ring  (see  a,  b,  c,  fig.  78).  The  initial  angle  i  is  already 
large  and  shows  the  rotation  of  compensator  from  the  vertical  needed  to 
bring  the  ellipses  into  the  field.  Two  sets  of  experiments  were  made  with 
plates  respectively  d  =  0.965  cm.  and  d  =  0.705  cm.  in  thickness,  with  the 
results  given  in  table  30. 

TABLE  30. 


Experiment  I. 

d 

j 

h 

Afc 

0.965  cm. 

9.0° 

IO.I 

11.4 

0.0529  cm. 
.0596 
.0676 

0.0067  cm. 
.0080 

Experiment  II. 

d 

i 

h 

Afc 

0.705  cm. 

2.6° 

3-3 
3-9 

o.oioocm. 
.0139 
.0165 

0.0039 
.0026 

The  equation  of  the  preceding  section  is  used  for  h.  The  first  case  shows 
about  the  same  order  of  sensitiveness  (M  per  half-ring).  In  the  second  case, 
for  the  thinner  plate,  the  sensitiveness  has  been  more  than  doubled.  This 

*  The  same  result  may  be  obtained  in  the  absence  of  the  compensator  by  rotating  N  and 
N'  on  a  horizontal  axis,  successively  by  small  amounts,  into  parallelism  with  M  and  M'. 


REVERSED  AND   NON-REVERSED   SPECTRA. 


117 


is  in  a  measure  not  unexpected,  because  the  amount  of  displacement,  caet. 
far.  (i.e.t  the  mobility  and  size  of  ellipses),  increases  in  marked  degree  as  the 
plate  compensator  is  thinner.  But  apart  from  this  AA  is  in  some  way,  yet 
to  be  stated,  associated  with  the  obliquity  of  rays  in  a  vertical  plane. 

In  case  of  the  rotating  compensator,  vertical  and  lateral  displacement  of 
centers  of  ellipses  go  together.  It  is  therefore  next  in  order  to  determine  the 
ratio  of  vertical  and  lateral  displacement. 

The  equation  deduced  in  §  8,  which  follows  easily  from  figure  77,  may  be 
put  in  the  form  (for  a  single  passage  of  light  through  the  plate) 

n\  =  2e(sw?i/2  — /*  sin2  r/a) 
or  into  the  approximate  form  for  small  angles 

n\=e(n—  i)f/2fji 
From  the  former  equation 

di  \ 


dn    0(sin  i — cos  i  tan  r) 


or  approximately,  again, 


di 

dn 


X/z 


Hence,  if  At  corresponds  to  x  fringes, 
di 


=  x-~r  =x  ~,  -  . 
dn        ei(ji—  i) 


roughly 


Again,  for  the  corresponding  normal  displacement  AN  of  the  micrometer  at 
the  opaque  mirror, 

#X  =  2AJVcos  6 
Hence 

__  ei(n  —  i)A*  _  aA/V  cos  6 
X/x  ~^~~ 

The  data  given  in  table  3  1  were  found  from  successive  positions  of  the  plate 
0  =  0.705  cm.,  while  the  center  of  ellipses  moved  as  in  figure  78,  at  the  D  line. 

TABLE  31. 


Center  at 

i 

At 

X 

loW 

io*AN 

X 

Bottom..   .. 

13-6° 

19  cm. 

... 

0  =  45° 

Middle..  .. 

12.5 

i.i 

18 

28 

9 

22 

x  =  i  .27  /A* 

Top  

«-s 

I.O 

15 

35 

7 

17 

x  =  24,oooA^V 

Bottom..   .. 

7-5 

in 

io«A  =  59;A«  =  i-53 

Middle..   .. 

6.7 

0.8 

7 

"5 

4 

10 

Top    .  . 

5-9 

0.8 

6 

119 

3 

8 

In  view  of  the  small  values  of  AAT  and  A*  and  the  estimated  n  and  B,  the 
two  sets  of  values  of  x  are  no  more  divergent  than  would  be  expected.    The 


118 


THE   INTERFEROMETRY  OF 


values  of  *  are  very  different  in  different  adjustments,  because  the  ellipses 
may  also  be  raised  and  lowered  by  rotating  one  of  the  opaque  mirrors,  as  M, 
around  a  horizontal  axis,  though  in  this  case  with  rapid  loss  of  sharpness. 
Here  the  rotation  of  both  mirrors  of  a  pair,  M'  and  M  for  instance,  by  the 
same  amount,  is  required  as  an  equivalent  to  the  rotation  of  the  compensator, 
as  has  been  stated.  The  aim  is  to  render  both  parallel  pairs  themselves  parallel. 
If  x\  is  eliminated  from  the  two  equations  on  the  preceding  page, 

i 
n=- 


The  above  data  are  inadequate  for  evaluating  n,  but  they  nevertheless 
indicate  a  value  entirely  too  low.  It  seems,  therefore,  as  if  some  essential 
term  has  been  left  out  of  sight.  This  is  also  to  be  inferred  from  the  values 
of  x,  which  differ  systematically  on  the  two  sides  of  the  table  31. 

TABLE  32. — Refraction  of  a  glass  plate,  e  =  0.434  cm. 
^  =  1.53.  Rotation  around  vertical  axis.  Jamin 
type  displacement  interferometer.  0  =  45°. 


i 

AATXio3 

At 

i 

AWXio3 

M 

cm. 

cm. 

+  12.5° 

2.60 

156 

-  6.5° 

.60 

10.5 

1.65 

147 

-  8.5 

1.  20 

"•56 

8-5 

1.25 

151 

-10.5 

i-75 

•52 

6-5 

.70 

-12.5 

2.60 

•56 

4-5 

•45 

-14-5 

3-35 

•51 

2-5 

•05 

-16.5 

4.60 

•57 

±    0.0 

.00 

-18.5 

5-6o 

•55 

-  4-5 

•30 

80°   yr   100'   w 


In  view  of  the  discordant  results  obtained  here  and  elsewhere  with  this  type 
of  rotating  compensator,  the  provisional  parts  of  the  apparatus  were  improved 
by  mounting  a  more  accurate  graduated  circle  with  vertical  axis  and  tangent 
screw.  A  good  plane-parallel  plate  of  thickness  e  =  0.434  cm.  and  refractive 
index  ^=1.53  was  then  adjusted  normal  to  the  ray  passing  through  it,  by 
noting  the  position  of  reversal  of  motion  of  the  ellipses  both  when  the  plate 
was  rotated  around  a  vertical  and  around  a  horizontal  axis.  The  data  of 
table  32  and  figure  79  were  found  while  the  plate  was  rotated  on  its  vertical 
axis  both  in  a  clockwise  and  counter-clockwise  direction  from  the  normal 
position.  In  the  former  case  (left  side  of  curve)  the  normal  position  showed 
the  same  constants  before  and  after.  In  the  latter  this  was  not  quite  the  case, 
the  micrometer  being  not  sufficiently  refined  for  such  purposes. 

The  results  for  JLC  were  computed  from  the  full  equation 


AATcos0=*sin2-Ui — 


They  are  as  good  as  the  small  values  of  AN  and  the  impossibility  of  obtaining 
the  zero  of  i  with  sufficient  sharpness  admit,  and  they  show  that  the  latter 
cause  adequately  explains  all  the  irregularities  encountered. 


REVERSED  AND   NON-REVERSED   SPECTRA. 


119 


The  equations  are  liable  to  be  cumbersome  in  the  cases  of  greatest  interest. 
I  therefore  proceeded  experimentally  to  obtain  a  limit  of  Ah  or  Ax  per  fringe, 
using  thinner  glass  compensators.  The  results  are  given  in  table  33.  The 
ellipses  were  now  so  large  and  distorted  that  it  was  difficult  to  define  the 
center  of  irregular  rings.  The  transverse  displacement  is  therefore  largely 
referred  to  the  top,  middle  and  bottom  of  the  spectrum  band,  which  took 
up  about  one-third  of  the  height  or  diameter  of  the  field  of  the  telescope. 
The  angular  width  of  the  latter  being  about  3°,  the  corresponding  angular  height 
of  the  spectrum  is  thus  about  1.0°.  In  view  of  the  large  rings,  moreover,  the 
displacement  AAT  at  the  micrometer  of  the  mirror  M'  is  difficult  to  obtain  and 
the  data  given  are  estimates.  Experiments  like  the  present  must  be  made 
with  optic  plate-glass,  so  that  sharp  rings  nearly  circular  may  be  obtained, 
if  the  data  are  to  be  quite  satisfactory.  In  table  33,  A/t  thus  corresponds  to 
a  transverse  displacement  of  one  component  ray  parallel  to  itself,  equivalent 
to  a  displacement  of  the  centers  of  ellipses  of  about  0.5°  at  the  sodium  line. 

TABLE  33. — Vertical  (transverse)  displacement  of  ellipses.  Glass-plate 
compensators  n  =  1.53.  Horizontal  axis.  Angle  of  telescopic  fields0; 
angular  height  of  spectrum  about  1.0°  or  0.175  radian.  Vertical 
diameter  of  first  fringe  in  excess  of  height  of  spectrum. 


Fringe  centers  at 

e 

f 

h 

A&Xio4 

A#Xio4 

Top  of  spectrum  
Middle  of  spectrum  .  . 
Bottom  of  spectrum.  . 
Bottom  of  spectrum.  . 
Middle  of  spectrum  .  . 
Top  of  spectrum  

cm. 
0.300 

.300 

0.4° 
'•9 

2.6 

2-3 
i.i 
o.o 

cm. 
0.0007 
34 
47 
.0042 
20 
oo 

cm. 
-26 

0 

+13 

+22 
00 
20 

cm. 

B  .. 

O-434 

40.8° 

0.1274 

+  15 

o 

M  

40.2 

.1259 

00 

7 

T  
B 

.4.  -74 

•K 

.1217 
.0213 

-42 
+  12 

22 
O 

M                        

7.6 

.0200 

00 

0 

T 

f  7.1 

.0187 

—  13 

2 

B 

•4^4 

6.5° 

.0170 

+29 

0 

M 

5-4 

.0141 

oo 

2 

T 

4.1 

.0112 

—29 

7 

B 

474. 

f  3.3 

.0087 

+  19 

o 

M 

2.6 

.0068 

oo 

4 

T 

1.5 

.0039 

—29 

4 

B  

.O2O 

20.0 

.00253 

+  14 

M 

9.4 

.00115 

00 

T  

-3-6 

-.00044 

-16 

The  mean  result  of  all  data  is  here  about  AA  =  0.002  cm.,  and  this  is  not 
influenced  in  a  discernible  way,  either  by  the  thickness  of  plate  e  or  by  the 
rotation  angle  of  the  compensator  i.  The  smallest  value  M  =  0.0012  cm. 
appears  incidentally  and  not  when  the  system  of  four  mirrors  is  most  nearly 
in  parallel  adjustment.  The  transverse  displacement  of  ellipses  changes  sign 
with  the  sign  of  the  rotation  of  i  in  all  cases  and  is  independent  of  the  normal 
position.  The  longitudinal  displacement  reverses  at  the  normal  position. 


120  THE   INTERFEROMETRY   OF 

63.  Broad  slit  interferences.  Achromatic  fringes. — Some  allusion  has  been 
made  above  to  a  type  of  interferences  totally  different  in  size  from  the  regular 
fringes  and  seen  in  the  broadened  slit.  These  were  finally  isolated  and  show 
exceedingly  interesting  properties.  They  appear  to  best  advantage,  in  the 
absence  of  the  spectroscope,  in  the  broad  white  field  of  a  very  wide  slit.  The 
latter  may  be  removed.  They  have  the  appearance  when  vertical  of  regular 
Young  or  Fresnellian  fringes,  very  sharp  and  fine,  achromatically  black  and 
white  at  the  middle  of  the  grid,  colored  and  fainter  outward.  They  are 
vertical  when  the  enormously  larger  spectrum  fringes  discussed  above  are 
centered.  Like  these,  they  partake  of  displacement  here  through  the  broad 
white  slit  image,  and  this  displacement  is  extremely  sensitive  in  relation  to 
the  displacement  of  the  opaque  mirror  M'  (fig.  73)  to  which  it  is  due.  Thus 
a  displacement  of  AAT=  10"*  cm.  of  the  latter  corresponds  to  a  march  of  fringes 
through  about  0.017  of  the  telescopic  field  of  3°;  i.e.,  to  0.05°.  This  comprises 
two  fringes  or  A/V  =  5Xio"6  cm.  per  fringe.  Now,  these  fringes  are  so 
sharp  and  luminous  that  it  should  be  possible  on  proper  magnification  to 
measure  a  few  hundredths  of  this  with  an  ocular  micrometer.  It  is  from  this 
point  of  view  that  I  regard  the  new  fringes  important.  They  supply  the  fine 
fiducial  mark  in  displacement  interferometry  for  which  I  have  long  been 
seeking.  They  appear  in  a  white  field,  thus  requiring  no  spectrum  resolution 
nor  monochromatic  light.  Moreover,  the  source  of  light  need  not  be  intense. 

To  have  a  distinctive  name  for  these  fringes  which  will  be  much  used  in  the 
work  following,  I  shall  refer  to  them  under  the  term  "  achromatic  fringes."  If 
not  too  large,  the  central  fringes  are  straight  and  almost  quite  black  and  white. 

The  displacement  of  fringes  with  AN"  at  the  mirror  (when  wX  =  2AJVcos  0) 
is  so  rapid  that  if  they  are  lost  it  is  difficult  to  find  them,  unless  the  centered 
large  spectrum  fringes  in  the  spectroscope  are  first  reestablished.  The  latter 
are  easily  found.  A  removal  of  the  prism  grating  g,  figure  73,  and  a  widening 
of  the  slit  show  the  achromatic  fringes.  The  datum  for  sensitiveness  may 
be  found  directly  as  follows:  The  displacement  at  the  mirrors  corresponds 
to  about  two  residual  fringes.  Thus  a  single  fringe  (distance  apart  of  the 
intensely  black  lines  at  the  center  which  can  be  distinguished  and  used  as 
fiducial  lines  for  this  very  reason)  corresponds  to  a  displacement  of  mirror 
of  AA/^=5oXio"6,  as  above.  The  white  pattern,  as  a  rule,  appears  but  once 
and  is  not  usually  present  rhythmically,  as  is  the  phenomenon  in  the  next 
section  for  homogeneous  light. 

As  a  clue  to  the  nature  of  the  residual  fringes,  one  may  note  in  the  first 
place  that  they  may  be  recovered  in  the  principal  focal  plane,  if  the  two  white 
slit  images  which  may  have  separated  be  put  in  coincidence.  Without  such 
coincidence  they  are  seen  sharply  in  other  focal  planes. 

Later,  on  more  careful  adjustment  as  to  parallelism  by  the  auxiliary  normal 
method  described  below,  periodic  reappearance  of  the  achromatic  fringes 
was  in  fact  obtained.  The  central  set  was  exceedingly  strong  and  sharp  as 
usual.  To  the  right  and  left  of  it  similar  patterns  or  groups  rapidly  decreasing 
in  strength  were  discovered.  Not  more  than  two  patterns  on  each  side  of  the 


REVERSED  AND   NON-REVERSED   SPECTRA. 


121 


central  set  could  be  seen.  The  white  field  between  the  patterns  was  several 
times  the  width  of  the  fringed  field.  Each  group  as  a  whole  resembles  the 
fringes  of  the  biprism,  as  usual,  and  they  differ  appreciably  only  in  intensity 
and  in  their  focal  planes.  It  is  difficult  to  account  for  this  periodic  reappear- 
ance; but  it  must  be  due  to  reflections  at  the  half -silver  surfaces.  The  reflec- 
tions from  the  uncovered  surfaces  are  indeed  just  visible,  and  naturally,  since 
they  are  duplicates,  they  also  carry  the  fringes.  But  they  are  easily  differ- 
entiated by  the  relative  faintness  of  field  and  have  nothing  to  do  with  the 
recurrences  in  question. 

Ordinary  daylight  is  quite  adequate  to  show  the  residual  fringes  in  the 
complete  absence  of  the  collimator.  They  are  superimposed  on  the  field  of 
view  (landscape,  etc.)  and  hence  will  subserve  other  purposes  than  are  here 
given.  When  the  adjustment  for  parallelism  is  not  sharp,  the  fringes  may 
often  be  found  strong  in  continuously  varying  focal  planes. 

64.  Wide  slit.  Homogeneous  light.  Sodium  flame.— A  further  clue  to 
the  nature  of  the  residual  fringes  will  be  obtained  when  white  light  is  replaced 
by  homogeneous  light.  A  strong,  large  sodium  flame  near  the  mirror  M, 
figure  73 ,  suffices.  The  fringes  now  appear  of  the  same 
size  in  yellow  light,  naturally  spread  over  a  much 
larger  area  of  field.  But  on  moving  the  mirror  M' 
(A/V  increasing  continually)  forward  very  gradually, 
the  homogeneous  fringes  alternately  vanish  and  reap- 
pear, each  time,  however,  enlarged  in  size  (nearly 
doubled  but  still  straight)  until  at  an  intermediate 
position  of  symmetry  enormous  round  ovals  cover  the 
yellow  field.  The  fringes  then  diminish  symmetrically 
in  the  same  way.  The  following  data  for  the  microm- 
eter position  corresponding  to  [the  clearest  demarca- 
tions of  fringes  are  illustrative.  At  least  six  periods 
(n)  are  easily  detected  on  each  side  of  the  ovals 
(«  =  o) .  Thus  (originally  small  fringes,  vertical,  increasing  in  size  to  huge  ovals) 

n  =6          5  4  3  2  i  o.etc. 

AJVXio3  =  o        49        90        127        171        214        243  cm.,  etc. 

These  intervals,  since  it  is  impossible  to  establish  the  maximum  states  of 
presence  or  absence  of  fringes  quite  sharply,  are  practically  equidistant,  as 
figure  80  indicates.    Thus  the  mean  period  of  reappearance  is  A7V= 0.042  cm.; 
or  a  path-difference  of  2AATcos  0=0.059  cm.;  or  a  shift  of  ray  paralle 
itself  (2AJVsin  6  =  0.059  cm.)  of  the  same  amount. 

The  reason  for  this  rhythm  can  only  be  the  two  wave-lengths  oi 
and  A  lines  of  the  sodium  flame,  originally  detected  in  the  colors  of  thin  pi 
by  Fizeau.    Hence  a  relatively  enormous  shift  of  micrometer  of  nearly  0.5  mm. 
is  equivalent  to  the  wave-length  interval  AX=6Xio-«  cm.,  or  AX/AJV  = 


122  THE  INTERFEROMETRY  OF 

icr8/42Xicr3  =1.4X10-*.    Treating  the  case  in  terms  of  the  interferences  of 
thin  plates  and  two  wave-lengths,  X  and  \+d\, 


AX    Aw 
wA= constant,    or 

while 


wX = constant,    or    —  =  —  =  -  for  each  period 
X       n     n 


X2 

wX=—  =  2 AN  cosO    or    AW=X2/2AXcos0 

since  (0  =  45°)  approximately, 

X=6oXio"6  cm.,      AX  =  6Xio~8  cm.,      cos  0  =  0.71,     A/V= 0.041  cm. 

agreeing  as  nearly  as  may  be  expected  with  the  experimental  datum.  The 
apparatus  thus  serves  incidentally  for  investigating  such  properties  of  spec- 
trum lines  as  Michelson  in  particular  has  detected.  Finally,  with  the  sodium 
arc  the  data  of  figure  81  were  found,  where  black  dots  denote  fringes,  open  cir- 
cles a  clear  yellow  field.  The  mean  trend  is  about  AAT= 0.043  cm-  per  period 
or  wave-length  gained.  White  fringes  coincided  with  n  =  2. 

With  white  light  the  interference  grid  does  not  usually  reappear  rhythmi- 
cally, nor  does  it  correspond  to  the  zero  period  of  figure  80 — i.e.,  to  the  ovals 
for  sodium  light.  It  was  exactly  of  the  size  of  the  fourth  period,  in  yellow 
light;  it  always  coincides  with  the  central  ellipses  of  the  spectroscope  as  stated, 
but  does  not  require  sharp  horizontal  and  virtual  coincidence  of  the  super- 
posed images.  The  reappearance  of  the  achromatic  fringes  obviously  depends 
on  conditions  different  from  the  case  of  sodium  light. 

In  a  flash  of  the  arc,  showing  many  sharp  spectrum  lines  in  all  colors,  each 
of  the  lines  gives  evidence  of  the  phenomenon — i.e.,  if  the  residual  fringes 
are  oblique,  each  such  line  is  strongly  helical  in  appearance. 

If  a  single  compensator  (i.e.,  a  glass  plate  in  one  interfering  beam)  is  used 
and  the  path-difference  annulled,  the  fringes  are  visible  again,  but  very  rapidly 
grow  smaller  with  the  increase  of  glass-path.  If  compensators  of  nearly  like 
thickness  and  glass  are  used  in  both  beams,  they  nearly  neutralize  each  other 
if  at  the  proper  angle  one  to  the  other.  But  there  is  almost  always  an  out- 
standing micrometeric  difference  in  thickness  (if  ordinary  glass  plate  is  used) 
of  great  importance  in  modifying  the  residual  phenomenon.  The  following 
experiments,  for  instance,  were  made  with  a  compensator  0.944  cm.  thick 
in  the  rear  and  0.958  cm.  thick  in  the  front  beam,  both  of  the  same  glass.  The 
half -silver  mirrors  were  0.7  cm.  thick.  The  wide-slit  experiments  are  supposed 
to  start  after  the  spectrum  ellipses  first  to  be  found  (fine  slit)  have  been  cen- 
tered. The  fringes  are  always  clear  and  very  sharp  when  the  white  slit  images 
accurately  coincide. 

With  the  compensators  A,  B  at  the  proper  angle  and  rotating  in  the  same 
direction  (fig.  82),  fringes  of  very  slight  enlargement  were  seen  with  the  lines 
nearly  vertical.  If  the  compensators  A,  B  were  rotated  at  a  proper  pace 
in  the  opposite  direction  to  each  other,  A,  B,  figure  83,  the  fringes  /'  and  /" 
grew  rapidly  smaller  and  turned  toward  the  horizontal.  Since  the  fringes 
are  large  circles  and  the  beams  here  rise  and  fall,  respectively,  while  a  greater 


REVERSED  AND   NON-REVERSED   SPECTRA.  123 

glass  thickness  is  introduced,  this  result  is  to  be  expected.    It  shows  the  im- 
portance of  the  differential  glass-path. 

The  preceding  thick  half-silver  mirrors  (0.7  cm.)  were  now  replaced  by 
thinner  half-silvers,  the  glass  plates  being  each  about  0.3  cm.  thick.  The 
fringes  after  being  found  had  not  appre- 
ciably changed.  Another  pair  of  half- 
silvers  of  the  same  thickness  was  then 
installed  with  like  results.  But  now, 


on  adding  the  compensators  (0.944  and  •,/    '      & 

-Z    -^ 


x  X 


0.958  cm.)  as  above,  a  marked  enlarge-  82 

ment  of  fringes  resulted.    Small  differ-  7*  X 

ential  thicknesses  must  here  have  been  '-T  J 

accidentally  compensated.     Opposite  rotations,  as  in  figure  83,  rapidly  pro- 

duced .very  fine,  nearly  horizontal  fringes  /'/".     Rotation  as  in  figure  82 

left  the  vertical  fringes  nearly  intact,  but  on  passing  from  the  position  AB  to 

A'B'  very  marked  enlargement  occurred,  as  follows: 

Mean  angle  of  glass  plates.  —  45°  o°  +45° 

Mean  angle  subtended  by 
one  fringe  in  the  telescope.         0.0015  rad.        0.008  rad.          0.0005  rad. 

The  compensator  plates  were  now  exchanged  and  the  fringes  found  after 
centering.  They  proved  to  be  very  much  smaller,  the  angle  subtended  in 
the  same  telescope  being  only  about  0.0002  radian.  The  preceding  accidental 
compensator  has  therefore  been  destroyed  by  exchange. 

On  passing  through  the  normal  position  of  one  plate  in  figure  82,  the  fringes 
usually  incline  toward  one  side  or  the  other.  Thus  there  can  be  little  doubt 
that  the  fringes  in  question  are  due  to  slight  difference  of  glass-path  or  ex- 
tremely sharp  glass-wedge  excess  in  one  or  the  other  component  beam.  In 
fact,  I  found  eventually  that  fringes  could  be  enlarged  by  rotating  the  proper 
compensator  around  a  vertical  axis.  Large  fringes  (up  to  0.002  radian  in  the 
given  telescope)  are  usually  colored  and  curved  and  not  so  available  as  smaller 
fringes  highly  magnified.  It  is  in  this  way  (double  rotation)  that  it  was  pos- 
sible to  make  the  white  fringes  coincide  in  order  with  the  ovals  of  the  fringes 
for  homogeneous  light  (n=o),  the  orders  met  with  above  being  the  n-2  and 
n  =  4.  The  fringes  resemble  those  of  Fresnel's  biprism;  but  as  they  are  seen 
with  a  wide  slit  or  in  the  absence  of  a  slit  only,  as  they  coincide  with  the  cen- 
tered spectrum  ellipses  of  a  fine  slit  and  as  they  are  a  definite  order  (second, 
for  instance)  of  the  fringes  seen  with  a  flame  of  homogeneous  light,  they  are 
necessarily  referable  to  the  colors  of  thin  plates. 

Hence  the  equation  for  these  fringes  may  be  assumed  to  be  (as  may  be 
seen  from  figure  84,  where  A  and  B  are  the  compensators) 

n\  =  (e-e')  (p  cos  (r  -  a)  -  cos  *) 

when  9  and  *'  are  the  thicknesses  of  the  two  half-silver  plates,  /*  their  index  of 
refraction,  i  the  angle  of  incidence,  r  the  angle  of  refraction  of  an  incident  ray, 
and  where  a  is  the  outstanding  angle  between  the  faces  of  the  differential 


124 


THE   INTERFEROMETRY   OF 


glass  wedge,  e— e'  thick  at  the  ray  in  question.  The  possibility  of  throwing 
these  fringes  into  any  order  of  size,  their  small  extent,  sharpness,  and  great 
abundance  of  light  constitute  their  value  for  measurement. 

Thus  it  is  furthermore  obvious  that  the  achromatic  fringes  must  also  be 
obtainable  in  Michelson's  interferometer,  in  any  order  of  size  and  in  a  field  of 
white  light.  Tests  were  made  with  this  object,  beginning  with  the  fine  slit 
and  the  centered  ellipses  of  the  spectrum  interferences.  Removing  the  spec- 
troscope and  enlarging  the  slit  indefinitely,  the  residual  fringes  appeared. 
They  were  never  so  strong  and  clear,  however,  in  any  order,  as  was  the  case 
with  the  Jamin  interferometer,  although  many  trials  with  different  compen- 
sators were  made.  They  would  not  be  useful  for  measurement.  Removing 
white  light  and  replacing  it  by  sodium  light,  the  white  fringes  were  found  to 
be  of  exceedingly  high  order,  more  than  0.7  cm.  of  micrometer-screw  being 
needed  before  the  circles  of  the  yellow  field  were  approached.  The  latter 
were  very  vague  and  usually  not  seen  in  the  principal  focal  plane  of  the  tele- 
scope. Since  the  rays  retrace  their  path  in  the  Michelson  interferometer,  the 
raising  and  lowering  of  the  spectrum  ellipses  is  not  possible;  but  the  residual 
fringes  may  be  put  in  any  order  by  changing  the  differential  glass-path  of  the 
rays.  They  appear  but  once,  not  rhythmically  like  the  sodium  fringes. 

The  equation  for  this  phenomenon  would  thus  be  (since  the  rays  retrace 
their  paths) 

«X  =  20ju  cos  (r  —  a) 

where  e  is  the  thickness  of  the  single  half -silver  and  a  its  effective  wedge  angle, 
positive  or  negative. 


/       ci        e:      ,sfi      } 


84 


To  obtain  the  circular  fringes  in  Michelson's  interferometer  with  a  wide 
slit  and  homogeneous  light,  the  rays  must  rigorously  retrace  their  path — 
i.e.,  all  reflection  must  take  place  at  the  same  spot  on  the  half -silver  plate. 
When  this  is  the  case  the  fringes,  even  though  obtained  with  common  plate- 
glass  and  a  sodium  flame,  are  beautifully  circular  and  sharp.  This  is  due  to 
the  fact  that  so  small  a  part  of  the  plate  is  used.  They  are  stationary  and 
exhibit  the  Fizean  periods  due  to  the  doublet  DiD2,  admirably.  With  white 
light  the  fringes  are  faint  and  useless.  When  the  rays  do  not  accurately 
retrace  their  paths — i.e.,  when  there  are  two  spots  of  light  on  the  half -silver 
one  or  more  centimeters  apart — the  fringes  are  soon  linear  and  very  small, 
as  above. 

With  regard  to  the  last  equation,  if  N  is  the  difference  of  normal  distances 
to  the  two  opaque  mirrors  (M,  N)  of  the  Michelson  interferometer,  from  the 


REVERSED   AND   NON-REVERSED   SPECTRA.  125 

two  respective  extremities  of  the  normal  to  the  half-silver,  at  the  point  where 
e  incident  ray  impinges  on  it,  the  equation  may  be  more  completely  written 


if  «  is  temporarily  disregarded.  N  is  independent  of  color  X,  but  otherwise 
represents  the  difference  of  air-paths  of  the  two  interfering  rays.  From  this 
equation 


_  _ 

dn      2*0  cos  R-  (X/cos  R)  (<fc/dX)  )  -2N 
so  that  the  center  of  ellipses  is  at  X  in  the  spectrum  when 

(3)  N=Nc=e(»  cos  R- 

\ 

Furthermore, 

(4)  ^=^_ 

dN     N-N, 


On  the  other  hand,  in  case  of  homogeneous  light  of  wave-length  X, 


_  =  _  _ 

dn      2n  cos  R.  (de/di)  -  20  tan  R  cos  *      n\(de/e)/di-  ze  tan  /?  cos  i 
where  cte/cfc  may  be  either  positive  or  negative. 

Centers  occur  when 
,£.  de      e  tan  R  cost  cfe 


—  =  —  p        or    —  = 

a*          /i  cos  ^?  <f  /? 

Thus  d»/dn  is  never  independent  of  X  and  the  centers  of  equation  (6)  are,  as  a 
rule,  quite  different  from  those  of  equation  (3).  If  the  angle  a  is  admitted, 
equation  (6)  takes  the  form 

(7)  ^  =  *tanK(i-2a/sin2jR) 

dr 

The  occurrence  of  the  residual  fringes  for  the  same  adjustment  of  the 
micrometer  as  the  centered  spectrum  fringes  is  thus  incidental.  To  obtain 
the  latter  the  two  superimposed  spectra  of  a  fine  slit  must  coincide  horizontally 
and  vertically  throughout  their  extent  —  i.e.,  the  two  linear  white  slit  images 
must  coincide.  If  now  the  slit  is  indefinitely  widened,  there  are  two  vertical 
lines  in  the  superposed  broad  white  images  which  are  completely  in  coinci- 
dence. In  case  of  ordinary  plate-glass  the  remainder  of  the  images  will  not 
be  mutually  in  coincidence  or  generally  there  can  not  be  coincidence  in  every 
color.  The  residual  or  achromatic  fringes  are  found  at  and  near  the  line  of 
coincidence  in  question.  Hence  if  either  opaque  mirror  is  slightly  rotated 
on  a  vertical  axis  the  residual  fringes  pass  from  edge  to  edge  of  the  broad 
white  slit  images,  S  and  5',  figure  85,  of  slightly  unequal  width.  Thus  if  the 
mean  breadth  is  5  and  the  difference  of  breadth  AS  =5'  -5,  the  very  small 


126  THE   INTERFEROMETRY   OF 

angle  corresponding  to  AS  is  measured  by  the  motion  of  the  fringes  over  the 
relatively  large  angle  5.  Hence  this  is  a  sensitive  method  for  measuring  angles 
which  will  be  utilized  below. 

Finally,  the  cause  of  displacement  is  to  be  given.  If  the  fringes  were  homo- 
geneous in  light,  they  would  fill  the  whole  field  and  simply  wander  indistin- 
guishably  to  or  from  the  center  of  homogeneous  circles  when  the  micrometer 
is  moved.  But  when  white  light  is  used  the  phenomenon  is  narrowed  to  a  few 
fringes  systematically  grouped  about  two  sharply  distinguishable  achromatic 
vertical  fringes  in  the  middle.  The  displacement  of  these  is  thus  accurately 
measurable  and  they  may  always  be  brought  back  to  the  field  of  the  telescope. 
Moreover,  the  distance  apart  of  two  black  fringes  must  correspond  to  the 
mean  wave-length  of  light  —  i.e.,  if  AAT  is  the  displacement  of  the  micrometer 
mirror  corresponding  to  a  fringe-breadth  for  the  angle  of  incidence  i  (here  45°), 

2  AN  cosi=\ 
or 


agreeing  reasonably  closely  with  the  above  rough  estimate  of  A/V  =  5  X  icr5  cm. 
If  we  take  the  equation  for  the  residual  fringes  in  the  Jamin  apparatus  as 

n\  =  t(fjL  cos  R  —  cos  i) 

where  e=0—e'  is  the  differential  thickness  of  the  compensators  and  the  residual 
angle  a  is  neglected  (fig.  84), 

di_=  _  X  _ 
dn      (n  cos  R—  cos  i)  (de/di-\-e  sin  R/n  cos  R) 

so  that  for  large  fringes  the  compensators  must  be  so  chosen  that  both  c  and 
de/di  may  be  small,  where  dt/di  may  be  either  positive  or  negative. 
The  last  equation  may  be  written 

di  =  i  _ 

dn~  (de/e   ,     sin  .R 


and  accounts  for  the  rapid  decrease  of  size  of  fringes  with  the  differential 
thickness  of  the  plate  compensators. 

65.  Vertical  displacement. — In  conclusion,  the  rise  and  fall  of  spectrum 
fringes  (i.e.,  the  transverse  motion  observed  and  utilized  when  a  compen- 
sator of  the  form  figure  77  rotates  on  a  horizontal  axis)  must  be  considered. 
This  method  was  used  above  for  centering  / 

the  spectrum  fringes.    Naturally  the  actual   &-  -  -? *l.^———~* 

motion  of  centers  is  obliquely  upward  or  -^  ?i— ==— -r-r-T3T~^ 1  .  JP 
downward,  unless  the  increase  of  glass-  ^  ** 

path  is  compensated  by  the  micrometer  at  the  opaque  mirror.  The  rays 
leaving  the  collimator  are  parallel  in  a  horizontal  plane  only.  They  are 
not  collimated  in  a  vertical  plane.  Hence  these  rays  intersect  at  the 


REVERSED  AND   NON-REVERSED  SPECTRA. 


127 


conjugate  focus  of  the  objective  of  the  collimator,  usually  somewhere 
between  the  mirrors  M  M'  and  N  N'  in  figure  73.  This  intersection  is  nearly 
in  a  horizontal  line,  owing  to  the  horizontal  width  of  the  collimated  beam. 
Hence  as  in  figure  86  there  are  two  virtual  linear  sources  of  light,  a  and  b,  nor- 
mal to  the  plane  of  the  diagram,  the  rays  from  which  may  be  treated  for 
practical  purposes  as  capable  of  interfering,  since  they  come  originally  from 
the  identical  slit  of  the  collimator.  The  effect  of  rotating  the  compensator 
(fig.  77)  on  a  horizontal  axis  is  thus  to  move  these  linear  sources,  a  and  b, 
through  each  other  vertically,  and  hence  their  distance  apart  may  be  called 
h,  where  if  d  is  the  compensator  thickness,  *,  r  angles  of  incidence  and  re- 
fraction at  the  compensator,  ju  its  index  of  refraction, 

h  =  d  (sin  i  —  cos  *  tan  K)  =  di  -  —  -  nearly 
M 

The  fringes  will  thus  be  larger  as  h  is  smaller  (fig.  86),  in  accordance  with 
the  equation 


where  x  is  the  distance  apart  for  the  distance  r  and  0  the  angle  between  two 
fringes  observed  in  the  telescope,  for  instance. 

Experiments  were  made  to  test  the  last  equation  by  attaching  an  ocular 
micrometer  to  the  telescope,  so  that  if  x  is  the  distance  and  r  the  length  of 
the  telescope,  6  is  given.  The  distance  between  fringes  in  the  same  part  of 
the  field  was  then  measured  from  different  angles  of  incidence  i  at  the  com- 
pensator. The  results  were,  if  X  =  6Xicr6  cm.,  r=ig.5  cm.  (table  34). 

TABLE  34. 


i 

AXio» 

(X/A)Xio» 

* 

(*/r)Xio« 

0° 

3-6° 
6.6° 
8.6° 

ocm. 

22 
40 
51 

0.0 

2.8 

1.5 

1.2 

(0.2)  cm. 
•05 
•03 
.025 

2.6 

i-5 
1-3 

These  results  for  $=\/h  and  B=x/r  may  therefore  be  considered  as  iden- 
tical, since  the  fringes  vary  in  size  within  the  field  of  the  telescope. 

Finally,  a  displacement  AA/"  at  the  opaque  mirror  will  move  the  virtual 
sources  a  and  6  in  a  horizontal  plane,  aAA/"  cos  /  in  the  direction  of  rays  and 
2AAf  sin  I  transverse  to  that  direction  if  /  is  the  angle  of  incidence.  Hence 
b  is  usually  f6und  at  some  point  c  and  moves  into  c'  by  the  rotation  of  the 
compensator  in  question  about  a  horizontal  axis.  The  fringes  do  not  there- 
fore necessarily  pass  through  infinite  size  unless  c  is  at  b,  which  would  then 
pass  through  a.  The  condition  of  maximum  sensitiveness  in  transverse 
displacement  is  therefore  a  large  fringe-angle  6,  a  condition  which  requires 
use  of  optic  plate.  Fringes  subtending  i°  would  admit  of  Afc  =  3-5X10-'  cm. 


128 


THE   INTERFEROMETRY   OF 


per  fringe,  and  that  is  about  the  mean  value  obtained  in  table  33  and  else- 
where. In  the  present  mode  of  treatment  merely  practical  conveniences  are 
aimed  at.  A  more  uniform  method  will  be  given  in  the  next  chapter. 

66.  Angular  displacement  of  fringes. — Having  for  other  purposes  installed 
an  ocular  micrometer  in  connection  with  the  telescope,  it  seemed  worth  while 
to  make  a  direct  test  of  the  equations  given  elsewhere.*  These  are  apart 
from  signs 

Ac  ,    „».«.     -  -  B     —f-     -  ~-  f~ 

~T~  I     T~T" 


d\ 


d\ 


When  n=A+B/\z,  \=D  (sin  i— sin  0),  and  when  two  plates,  the  half-silver 
of  thickness  e  and  angles  of  incidence  *  (constant)  and  of  refraction  R,  and 
a  compensator  of  thickness  E  and  at  normal  incidence  are  included,  these  may 
be  changed  to 

'dNre=(2B/\*)  Dcos' el     e     /       zB  tan2^"^1 

(cos#V  X2  M  / 
Here  dd/dNe  is  the  displacement  of  the  center  of  ellipses  per  centimeter  of 
displacement  of  the  normal  micrometer  at  one  opaque  mirror  of  the  Michel- 
son  device,  for  the  wave-length  X.  D  is  the  grating  constant  and  0  the  angle  of 
diffraction.  In  the  apparatus  used  JD  =  i67Xio~6  cm.,  0  =  2o°4o'  at  D  line, 
*  =  3o°,  ^=19°  5'  in  case  of  glass  and  17°  52'  in  case  of  carbon-bisulphide 
plates;  X=  58.93X10-*  cm.;  £=4.6Xio~u  or  2#/X2=o.o265,  M=i-53  (glass). 
Under  these  circumstances  the  term 


and  may  be  neglected  in  comparison  with  3.    Thus  the  equation  takes  the 
simpler  form 

d6_ X i__ 

dNe~  (6B/X2)  D  cos  0  E+e/cos  R 

The  values  of  E  and  e  are  given  in  table  35.    The  data  marked  dd/dNe  (com- 
puted) in  the  table  were  found  from  the  last  equation. 

TAELE  35. — Reduction  of  sensitiveness  by  glass  thickness.  D  =  d/cos  r.  Incidence  at 
t  =  30°,  R  =  ig°  5'  (glass),  17°  52'  (CSj),  compensators  normal.  Glass:  M  =  i-53;  CS2, 
Ai  =  i.63.  Telescope  19.5  cm.  long.  25/Xz  =  o.O26s.  0  =  167X10'*  cm.  0D  =  2O°  40'. 


Detail. 

e 

E 

Glass. 
ed/cos  R 

CS, 
E/cosR 

Total 
E'  ±E 
+«/cos  R 

dx 
dNe 

do 

dNe 
observed. 

de 
dNc 
com- 

puted. 

cm. 

cm. 

cm. 

cm. 

cm. 

Glass  -j-CS,... 
Glass  -f  glass 
Glass  
Glass—  glass 
Glass—  glass 

+0.27t00.27 

0.695 
.695 
•695 
•695 

0.60 
.562 

.0 

.247 
.562 

.0 
•735 

•735 
•735 
•735 

0.63 
.562 
.0 
.247 
.562 

0.63 
1.297 
•735 
.488 
•173 

33 

182 

500 

1.70 

r5 

9-3 

25.64 

J| 

*  Carnegie  Inst.  Wash.  Pub.  No.  229,  1915,  p.  74  et  seq.,  \\  40,  41.    In  equations  (13) 
md  (18)  d\/dNe  and  dd/dNc  should  be  inverted  and  D  cos  e  in  the  latter  put  in  the  numerator. 


REVERSED  AND   NON-REVERSED   SPECTRA. 


129 


If  %  is  the  displacement  in  centimeters  in  the  ocular,  0=x/ig.5,  the  denom- 
inator being  the  length  of  the  telescope.  The  successive  values  of  *  in  terms 
of  N,  the  mirror  displacement,  are  given  graphically  in  figure  87  for  the  dif- 
ferent values  of  £'=£+*/cos  R  in  question,  and  from  them  de/dNt  was 
taken  graphically.  All  the  curves  are  appreciably  straight,  except  the  one 
for  £'=1.3  cm.,  which  is  definitely  curved. 


The  values  of  dd/dN  e  observed  and  computed  obviously  agree  as  closely 
as  may  be  expected  for  mean  conditions  of  the  spectrum  between  green  and 
red  and  of  the  mode  of  procedure.  The  small  coefficient  for  carbon  bisul- 
phide is,  moreover,  in  keeping  with  its  large  value  of  B  or  high  dispersion. 
In  a  sensitive  displacement  interferometer  B  like  E'  D  should  all  be  small; 
but  unfortunately  this  implies  large  ellipses.  The  beautiful  small  ellipses 
obtained  with  the  carbon-bisulphide  plate  are  of  little  avail  because  of  their 
sluggish  motion.  In  fact,  the  small  thickness  of  the  carbon  bisulphide  layer 
and  the  reduced  coefficient  observed  are  quite  striking  in  comparison  with 
glass.  Moreover,  since  for  an  order  n 


dn       2  D  cos  6  (cos  R+zB/X-  cos  R)  -N 

\he  effect  of  a  large  B,  though  it  reduces  the  general  size  of  ellipses,  does  so 
but  slightly,  since  the  term  in  B  can  not  be  more  than  a  few  per  cent  (3  to  6) 
of  the  other  term  in  the  binomial. 


CHAPTER  VI. 


THE  DISPLACEMENT  INTERFEROMETRY  OF  SMALL  ANGLES  AND  OF  LONG 
DISTANCES.     COMPLEMENTARY  FRINGES. 

67.  Parallel  rays  retracing  their  path.  —  The  following  method  was  devised 
with  a  view  to  the  micrometric  measurement  of  angles.  It  will  be  used  else- 
where in  connection  with  an  electrometer  for  reading  microvolts.  An  inter- 
ference method  of  a  different  kind  for  measuring  small  angles  was  developed 
some  time  since  and  used  at  length  in  connection  with  the  deviation  of  the 
horizontal  pendulum.*  Again,  the 
electrometer  was  treated  in  different 
wayst  by  the  aid  of  the  interferom- 
eter. The  present  method,  however, 
will  differ  from  all  of  these.  In  figure 
88,  L  is  a  horizontal  beam  of  white 
light  from  a  collimator  after  passing 
through  the  auxiliary  clear  plate  P 
(to  be  used  preliminarily  for  paral- 
lelizing the  mirrors  of  the  system 


in   a  way  presently  to  be  shown), 


88 


the  beam  is  reflected  at  a  and  b  by  the  half-silver  plates  Hi  and  Ht  respect- 
ively, to  the  wide  opaque  mirror  m.  The  rays  now  retrace  their  paths 
or  nearly  so,  to  be  in  turn  transmitted  at  a  and  b  by  the  half-silvers 
HI  and  Hz.  These  transmitted  pencils  similarly  impinge  on  the  opaque  mir- 
ror M  and  the  half-silver  Hs  at  c  and  d  respectively,  and  pass  thence  (the  ray 
from  c  being  transmitted)  into  the  telescope  at  T.  The  direct-vision  grating 
prism  g  may  be  swiveled  in  place  or  removed  at  pleasure. 

To  bring  the  system  of  four  mirrors  into  complete  parallelism  is  here  of 
considerable  importance  if  the  spectrum  fringes  or  the  achromatic  phenomenon 
are  to  be  adequately  large  for  measurement.  The  presence  of  the  common 
mirror  m,  however,  suggests  the  procedure.  When  the  clear  plate  P  is  in 
place,  the  rays  ae  and  bf  on  returning  are  also  again  reflected  at  a  and  b  toward 
L  and  may  be  clearly  seen  in  a  telescope  at  p.  Hence  if  m  is  the  standard 
plane  and  nearly  vertical,  the  mirrors  Hi  and  H2  will  be  parallel  when  the 
slit  images  seen  at  p  coincide  horizontally  and  vertically,  while  Hi,  Hz,  and  m 
will  have  their  common  normal  plane  in  the  diagram.  In  the  same  way  the 
mirrors  M  and  HZ  may  be  parallelized  with  their  common  normal  plane  in 
the  diagram.  Again,  the  return  rays  aL  and  bL  may  be  projected  on  the 
objective  of  the  collimator,  or  on  a  small  screen  near  it,  by  correspondingly 
focusing  the  collimator.  The  two  sharp  slit  images  are  put  in  coincidence 


*  Carnegie  Inst.  Wash.  Pub.  No.  229,  §  19  et  seq.,  1915. 
130 


t  Ibid.,  §  67  et  seq. 


INTERFEROMETRY  OF   SPECTRA.  131 

horizontally  and  vertically.  This  is  usually  more  convenient  and  as  a  rule 
adequate.  Other  methods  are  given  below. 

If  the  distances  ac  and  bd,  ab  and  cd  have  previously  been  made  nearly 
equal  and  the  angles  approximately  90°,  the  fringes  will  usually  be  found  on 
moving  the  micrometer-screw  normal  to  Hz. 

As  the  mirrors  are  thick  glass  plates,  it  is  preferable  that  the  half-silvered 
sides  of  Hi  and  H2  be  toward  L  and  the  half-silvered  side  of  H3  toward  T. 
In  this  case  each  ray  passes  the  plates  twice,  as  indicated  in  figure  88.  With 
ordinary  plate-glass  the  fringes  when  found  are  still  apt  to  be  small.  They 
are  then  to  be  enlarged  and  centered,  by  compensator  of  clear  glass  C  and  C, 
in  the  two  rays  respectively,  rotated  in  opposite  directions  around  a  hori- 
zontal axis  until  the  center  of  ellipses  is  in  the  field  of  the  spectroscope.  It 
may  be  necessary  to  actuate  the  micrometer-screw  at  d  to  complete  the  ad- 
justment. If  m  is  adjustable  on  two  axes,  the  compensators  C,  C'  are  super- 
fluous, as  will  presently  appear. 

When  the  ellipses  are  centered,  the  direct-vision  spectroscope  g  removed, 
and  the  slit  widened  or  removed,  the  residual  or  achromatic  fringes  appear  in 
sight  and  are  ready  for  use.  These  are  always  strong.  The  spectrum  fringes 
are  apt  to  be  less  so,  since  the  parts  of  the  ray  L  pass  through  two  half- 
silvered  surfaces,  HI  HZ  or  HI  H3,  in  succession.  The  spectrum  fringes  are  only 
sharp  when  the  slit  is  fine.  If  the  white  residual  fringes  are  too  dazzling,  a 
single  or  two  half-silvers  may  be  placed  before  the  objective  of  the  telescope 
with  advantage.  Two  plates  with  their  half -silvered  sides  in  contact  and 
held  so  by  a  steel  clip  are  excellent  for  this  purpose,  while  they  are  at  the  same 
time  protected  from  sulphur  corrosion.  This,  in  fact,  is  the  best  method  of 
preserving  silver  mirrors  (in  pairs)  when  not  in  use. 

If  a  is  the  fraction  of  light  transmitted  and  i  -  a  reflected,  the  fraction  of 
the  original  light  L  reaching  the  telescope  T  will  be  20^(1  —  a)2.  This  is  a 
maximum  if  a  =  y£.  Thus  the  illumination  is  reduced  to  jHj. 

When  the  spectra  are  in  coincidence  and  the  fringes  sharp,  the  mirror  m 
may  be  rotated  around  a  vertical  axis  at  A  into  some  position,  m'.  In  such 
a  case  the  two  spectra  will  move  through  the  field  of  the  telescope  at  T,  but 
their  coincidence  will  not  be  destroyed.  The  D  lines,  for  instance,  will  con- 
tinue to  be  superposed  throughout.  Considerable  path-difference  is,  however, 
introduced  in  this  way,  and  hence  the  fringes  will  march  through  the  spec- 
trum at  an  enormously  more  rapid  rate.  The  following  data  may  be  given, 
where  a  is  the  angle  of  rotation  of  the  mirror  m  and  N  the  reading  of  the 
micrometer  at  H3  (screw  in  the  normal  dri)  necessary  to  bring  the  center  of 
ellipses  back  to  the  sodium  lines.  In  both  cases  the  centers  were  out  of  the  field 
above  or  below,  so  that  horizontal  fringes  were  made  the  criterion  for  adjustment. 
This  method  is  somewhat  rough,  but  adequate  for  the  present  purposes. 

(i)  Fine  thin  fringes.    Relatively  large  glass-path.     Distance  ab,  figure  88,  *R-2i  cm. 
Thickness  of  glass  plates  (half-silvers),  e  =0.70  cm. 

«  =0°        0.05°        0.20°        0.30°        0.40°        0.50°        0.60° 

3          30  128  162  215          258          299  cm. 


132 


THE   INTERFEROMETRY   OF 


This  is  curve  a  in  figure  89.    From  it  the  mean  rate 

A2V 

-  =0.47  cm./degree,  or  27  cm./radian 
Aa 

may  be  found. 

(2)  Coarse  large  fringes.    Smaller  differential  glass-path. 
a=  o°   0.1°  0.2°  0.3°  0.4°  0.5°  0.6°  0.7°  0.8°   0.9°    1.0° 

-—  25     +29    84     134     176    217  265    323    365    420    467  cm. 


This  is  the  curve  given  (with  double  ordinates  for  distinction)  in  curve  b, 
figure  89,  and  in  figure  90.  Besides  this  the  datum  a=  —0.6°,  N=  —0.320  cm. 
was  obtained.  In  figure  90  the  mean  rate  is 

AN 

•  -  =0.465  cm./degree,  or  26.6  cm./radian 

Aa 

agreeing  with  the  preceding  as  closely  as  may  be  expected.  We  may  thus 
estimate  AAT  =  27X10-*  of  displacement  at  the  micrometer  at  H3  per  micro- 
radian  of  turn  a  at  the  mirror  m,  which  amounts  to  a  little  less  than  one 
interference  ring  per  micro-radian  (about  0.2  second  of  arc)  of  turn.  More- 
over, turns  of  a  less  than  a  few  degrees  are  certainly  measurable.  Through- 


Or 


5°      -6° 


92 


20em 


out  all  this  work  the  achromatic  fringes  are  also  available  for  precision  in  N, 
but  for  this  reason  are  more  difficult  to  manipulate  if  individual  fringes  are 
treated.  They  may,  moreover,  be  much  enlarged  by  rotating  the  mirror  m 
and  advancing  the  micrometer  at  HI  in  small  steps  in  such  a  way  as  to  pro- 
duce contrary  effects  and  thus  keep  the  achromatic  fringes  in  the  field.  If 
the  fringes  leave  the  principal  focus,  the  micrometer  at  HZ  and  its  adjust- 
ment screws  may  be  actuated  together  in  the  same  way.  This  is  the  most 
available  method  for  eliminating  the  glass-path,  so  that  enormous  spectrum 
ellipses  are  obtainable.  Finally,  three  groups  of  achromatic  fringes  (on 
each  side  of  the  strong  central  group)  were  noticed,  the  distance  apart  of 
groups  corresponding  to  about  AA/"= 0.0014  cm. 


REVERSED  AND   NON-REVERSED  SPECTRA.  133 

It  is  obvious  that  2A/V  cos  i  will  increase  with  zR,  the  distance  apart  of 
the  parallel  rays  ae  and  bf,  figure  88,  as  well  as  with  the  rotation  a.  But  the 
relations  are  not  obvious,  and  experiments  were  therefore  made  for  the  rela- 
tion of  AN  and  a  in  case  of  different  distances  apart  (2^  =  10,  21,  25,  28  cm.) 
of  the  rays  in  question.  The  results  are  given  in  figure  91,  where  the  abbre- 
viation r  =  AW/Aa.  In  figure  92  and  table  35,  furthermore,  this  value,  as 
obtained  from  figure  91  graphically,  is  compared  with  2R.  The  results  are 
not  quite  smooth;  but  as  zR  is  the  distance  apart  of  two  spots  of  light,  it  can 
not  be  specified  sharply.  Moreover,  a  variety  of  other  discrepancies  of  ad- 
justment enter  which  need  not  be  detailed  here.  From  figure  92  it  appears 
that  on  the  average  r/zR  =  0.024  in  terms  of  degrees  or  =1.35  in  terms  of 
radian.  Hence  we  may  consider  an  equation  of  the  form 


So  computed  cos*  would  be  0.73  instead  of  0.71,  but  the  difference  is  refer- 
able to  the  outstanding  glass-path  and  outstanding  air-paths  which  have  not 
been  included. 

68.  Groups  of  achromatic  fringes. — With  the  good  adjustment  stated,  a 
number  of  further  experimental  measurements  of  the  mean  position  (AAO  of 
the  fringe  patterns  on  the  micrometer  at  Ha  were  made,  with  results  as  fol- 
lows: The  size  of  the  individual  fringes  was  about  0.0225/35  =  0.00064  radian 
in  the  telescope  at  T — i.e.,  about  0.034°  of  arc.  The  fringe  patterns  will  be 
designated  primary,  secondary,  etc.  The  data  are  io3AAT  cm. 

Tertiary +2.6  2.7  2.7  cm.  XiQ-' 

Secondary +1.3  1.2  1.3         1.4  1.4 

Primary ±0.0  o.o  o.o        o.o  o.o 

Secondary —1.3  1. 1  ...         1.2  1.2 

One  group  was  visible  on  one  side  and  two  on  the  other  and  one  or  more 
frequently  escaped  capture  by  the  micrometer.  The  wave-length  difference 
of  these  positions  is  exceedingly  small.  Using  the  above  expression,  AX  = 
X2/2AAT  cos  6, 

AX  = 36Xl°"10 =!.9Xio^cm.  nearly 

2Xi.3Xio-3Xo.7i 

But  there  is  no  suggestion  in  this  datum.  The  displacement  AN  is 
equivalent  to  a  glass  thickness  e,  where  £#=*(/*- 1),  roughly.  Hence  e= 
i.3Xio-3/53=o.oo25  cm.  This  precludes  the  possibility  of  reflections  from 
the  two  sides  of  the  half-silver  film,  since  this  is  about  1,000  times  thinner, 
apart  from  its  index  of  refraction. 

Some  experiments  were  also  made  with  the   primary  achromatic  f 
(treated  as  a  single  group  and  not  individually,  as  the  angle  of  measurement 
of  a  was  not  correspondingly  delicate)   to  determine  Atf/A*    Two  series 
with  different  adjustments  showed  results  as  given  in  table  35  (2^-21  cm.;. 


134 


THE   INTERFEROMETRY   OF 


TABLE  36. 


Fringes  — 

Series  I. 

Series  II. 

a 

NXio* 

a 

NXio* 

Smaller  
Smaller 

—  O.2 
—    .1 
±    .0 
+    .1 
+    .2 

+  -3 

112 

47 
o 
56 
in 
170 

—0.2 
—   .1 
±    .0 

H4 
63 

0 

Very  large  
Smaller  

Smaller  

Smallest  

These  data  contain  the  new  result  that  the  achromatic  fringes  decrease  in 
size  with  great  rapidity  in  both  directions  of  the  increments  from  the  cen- 
tral position  (a  =  o).  Two  sets  were  usually  present.  The  rate  of  growth  as 
well  as  the  ratio  A/V/Aa:  is  not  the  same  on  both  sides  of  the  position  of  maxi- 
mum size  («=o).  The  mean  coefficient  may  be  found  graphically  as 

AAT 

—  =  0.56  cm./degree  or  32  cm./radian 
A  a 

which  is  larger  than  the  preceding  estimate  for  spectrum  fringes. 

This  experiment  bears  directly  on  the  nature  of  the  achromatic  fringes. 
It  shows,  moreover,  that  the  compensators  C  and  C'  in  figure  88  are  not 
essential  and  that  ellipses  may  be  centered 
by  rotating  the  mirror  m  on  a  horizontal 
and  vertical  axis.  It  is,  in  fact,  possible 
to  increase  the  achromatic  fringes  indefi- 
nitely in  size  in  this  way  ;  but  with  ordinary 
glass  they  eventually  become  sinuous  and 
no  longer  useful.  When  the  slit  is  fine 
and  the  ocular  out  of  focus,  well-marked 
hyperbolic  patterns  may  be  recognized;  but  these  become  straight  on  return- 
ing to  the  wide  slit  in  focus. 

69.  Measurement  of  small  angles  without  auxiliary  mirror.  —  This  method 
makes  use  of  the  original  apparatus,  figure  73,  but  the  two  mirrors  M  and  M' 
or  N  and  N',  figure  93,  are  rotated  together  as  a  rigid  system  around  a  ver- 
tical axis,  at  A,  for  instance.  In  view  of  the  absence  of  auxiliary  reflection, 
the  method  will  be  but  half  as  sensitive  as  the  preceding  one,  so  that  the 
equation 

AN  cosi 


is  sufficient  to  express  the  results.  But  on  the  other  hand,  if  spectrum  fringes 
are  to  be  observed,  there  is  greater  abundance  of  light,  since  the  half-silver 
film  is  penetrated  but  once  by  each  component,  ac  or  bd.  When  the  achro- 
matic fringes  are  used  the  light  is  always  superabundant  and  must  be  reduced 
in  intensity.  To  try  this  method  the  mirrors  N  and  N'  were  mounted  on  a 
good  divided  circle  so  as  to  rotate  together  on  a  rigid  arm  over  a  small  angle  a. 
The  achromatic  fringes  displaced  in  this  way  were  restored  by  advancing 


REVERSED  AND   NON-REVERSED   SPECTRA. 


135 


the  mirror  M'  over  the  distance  AAT  along  the  normal  micrometer-screw 
at  *'.  The  following  is  an  example  of  the  results  of  corresponding  values  of 
AN  and  Aa,  when  the  distance  apart  of  the  rays  ac  and  bd  was  2R  =  7  cm. : 

a  =0.0°    0.1°    0.2°       0.3°     0.4°      0.5°      0.6°     0.7°      0.8°     0.9°      10° 

AtfXio»  =  o.o  8.0  16.6  25.6  34.9  42.7  50.7  61.7  70.5  78.5  90.6  cm. 
These  results  as  a  whole  are  much  smoother  than  above,  for  incidental  rea- 
sons. From  a  graphic  construction  the  mean  rate  AAT/Aa= 0.088  cm./degree 
or  5.0  cm./radian  may  be  obtained.  Hence,  since  2^  =  7  cm.  (AAT/Aa)/2R 
=0.013  in  terms  of  degrees  or  0.72  in  terms  of  radians  (table  37).  This  re- 
sult is  roughly  half  the  preceding,  if  incidental  errors  be  disregarded.  From 
the  above  equation,  since  1  =  45°  nearly, 

AN       2R        7.0 

; —  = :  = =5-o  cm./radian 

Aa     2  cost     1.41 

agrees  closely  with  the  experimental  result. 

TABLE  37.— Values  of  r=AN/A «  micrometer  displacement 
per  degree  of  mirror  rotation. 


A.  Case  of  auxiliary  mirror,  fig.  88. 

Series 
No. 

AN 
Aa 

AN 
Aa 

2R 

¥1* 

Act  1 

6,7;  1,2 
8 
9 

10,  II 

cm./° 
0.47 

[26 

cm./rad. 
27 
33 
39 
15 

cm. 

21 

25 
28 
10 

mean. 
0.0241/  degree 
I-351/  radian 

B.  Case  of  rotating  pairs  of  mirrors,  fig.  93. 

I 

0.088 

5-o 

7 

o.o  1  261  /degree 
•72Vradian 

70.  Complementary  fringes. — Some  additional  attention  was  now  given  to 
the  hyperbolic  fringes  of  a  fine  slit  and  white  light  observed  when  the  ocular 
is  drawn  outward  or  to  the  rear  of  the  position  for  the  principal  focus  and  the 
spectroscope  is  removed.  They  appear  and  widen  with  the  washed  image  of 
the  slit.  They  are  quite  strong,  sharp  throughout,  and  gorgeously  colored, 
the  fields  and  shades,  figure  94,  a,  being  nearly  complementary  in  color.  The 
spectrum  fringes  must  be  centered  if  the  others  are  to  occur.  The  former, 
figure  94,  b,  are  usually  long  ellipses  or  hyperbolic  with  their  major  axes  horizon- 
tal, while  the  corresponding  new  fringes  are  hyperbolic  with  the  major  axes 
vertical.  They  are  extremely  sensitive  to  rotation  of  the  micrometer  mirror 
about  a  horizontal  axis,  rising  or  falling,  but  they  soon  vanish.  When  the 
micrometer  mirror  is  rotated  around  a  vertical  axis,  an  operation  which  sepa- 
rates the  white  slit  images  if  originally  coincident,  the  new  fringes  move 
bodily  by  displacement  from  left  to  right,  or  the  reverse,  depending  on  the 
sign  of  the  rotation,  while  they  continually  change  their  color-scheme.  When 
the  design  is  thus  displaced  as  a  whole  the  individual  fringes  move  as  shown 


136  THE   INTERFEROMETRY  OF 

in  figure  94,  a.  As  a  group  the  fringes  closely  resemble  the  lemniscates  of  a 
binaxial  crystal  in  polarized  light.  The  variation  of  the  color-scheme  is  probably 
the  same,  since  with  sodium  homogeneous  light  the  design  is  in  yellow  and 
black.  The  pattern  is  not  quite  dichroic,  but  appears  so,  red-green,  blue- 
yellow  combinations  with  an  intermediate  violet-yellowish  succeeding  each 
other.  In  polarized  light  the  figures  are  sharpened  as  a  whole,  but  there  is 
no  discrimination.  The  pattern  gradually  vanishes  with  a  wide  slit,  where- 
upon the  achromatic  fringes  may  be  seen  when  the  ocular  is  restored  to  the 
principal  focal  plane. 

If  the  white  slit  images  pass  through  each  other  (in  consequence  of  the 
vertical  rotation  specified)  the  direction  of  fringes  twice  changes  sign  in  rapid 
succession,  and  this  probably  occurs  when  the  white  slit  images  are  coin- 
cident. Barring  this  inversion,  the  march  is  regularly  proportional  to  the 
rotation. 

j|  With  the  displacement  (AN)  of  the  mirror  on  the  micrometer-screw  nor- 
mal to  its  face,  the  fringes  pass  through  a  continuous  succession  of  color- 
schemes,  but  soon  vanish,  for  they  coincide  in  adjustment  with  the  centered 
spectrum  fringes.  Similarly,  if  a  pair  of  mirrors  (MM'  or  NN',  fig.  93) 
rotates  about  a  vertical  axis  as  a  rigid  system,  the  same  continuous  change 
of  color-scheme  and  evanescence  is  apparent. 

These  interferences  differ,  naturally,  from  the  spectrum  interferences; 
they  also  differ  from  the  achromatic  interferences,  which  are  much  finer 
fringes,  partaking  of  the  regular  fringe  pattern  seen  with  biprisms.  They  are 
a  separate  phenomenon,  quite  sharp  and  definite,  occurring  under  like  con- 
ditions of  adjustment,  but  under  different  conditions  of  observation  (ocular 
out  of  focus  and  fine  slit).  In  the  principal  focus  the  two  sharp,  extremely 
bright  slit  images  are  alone  present.  They  are  absolutely  identical  in  struct- 
ure, however,  and  their  spectra  when  superposed  would  interfere  symmetri- 
ally  throughout  their  extent.  Under  these  circumstances  the  rays  intersect- 
ing in  the  white  slit  images  also  interfere  before  and  behind  the  principal 
focal  plane  of  the  telescopic  images  specified,  and  this  interference  is  not 
destroyed  when  the  slit  images  are  separated  (rotation  of  opaque  mirror 
about  vertical  axis),  or  when  the  slit  images  are  passed  through  each  other. 
What  is  not  easily  seen,  however,  is  the  reason  of  the  occurrence  of  large, 
sharp,  definite  hyperbolic  forms  instead  of  the  usual  Young  or  Fresnel  fringes 
of  two  slits  or  slit  images. 

On  the  Michelson  interferometer  these  fringes,  like  the  achromatic  fringes, 
are  extremely  faint  and  can  hardly  be  detected  except  by  putting  them  in 
slow  motion.  The  spectrum  fringes  are  equally  strong  in  all  cases.  It  appears, 
therefore,  that  the  two  half-silvers  are  favorable  to  evolving  both  the  hyper- 
bolic and  the  achromatic  sets  of  fringes.  The  Michelson  design  is  thus  not 
useful  here,  nor  for  the  measurement  of  small  angles  of  rotation  by  the  meth- 
ods described,  as  the  mirrors  would  have  to  be  rotated  in  opposite  directions. 

Further  work  with  the  complementary  fringes  on  different  interferometers 
of  the  Jamin  type  showed  that  to  produce  the  hyperbolics  the  fine  slit  images 


REVERSED  AND   NON-REVERSED   SPECTRA.  137 

must  coincide  horizontally  and  vertically.  They  do  not  in  this  case  probably 
coincide  in  the  fore-and-aft  direction,  for  the  plates,  etc.,  are  not  optically 
flat.  When  the  slit  images  are  separated  at  the  same  horizontal  level  into 
two  fine  parallel  lines,  the  complementary  fringes  in  fact  become  Fresnellian 
fringes,  finer  as  the  slit  images  are  more  separated  and  as  the  ocular  is  more 
rearward  or  forward.  This  is  precisely  what  should  occur.  We  may  conclude, 
therefore,  that  the  complementary  fringes  are  Fresnellian  interferences  of  two 
slit  images  and  that  the  central  hyperbolic  forms  are  due  to  outstanding  front 
and  rear  positions  of  the  two  slit  images,  which  seem  to  coincide  in  the  field 
of  the  telescope.  Differentiated  from  these,  the  achromatic  fringes  are  refer- 
able to  the  colors  of  thin  plates.  I  have,  in  fact,  also  succeeded  in  obtaining 
the  complementary  fringes  in  the  shape  of  broad,  straight,  gorgeously  colored 
vertical  bars,  without  suggestion  of  hyperbolic  contour. 


An  attempt  was  made  to  get  quantitative  estimates  of  the  passage  of  fringes 
on  rotating  the  paired  mirrors  over  an  angle  a.  To  control  the  small  angles 
the  device,  figure  95,  was  improvised  and  did  good  service.  Here  e  is  the 
tangent  screw  of  a  divided  circle  6  inches  in  diameter.  It  is  surrounded  by 
a  snug  annulus  of  cork  and  holds  the  brass  ring  /,  on  whose  surface  a  coarse 
screw-thread  has  been  cut.  Near  this  and  with  its  axis  in  parallel  is  a  quarter- 
inch  screw  a  and  socket  (not  shown)  controlled  by  the  disk  b.  A  strong  linen 
thread  c  is  looped  once  around  /  in  the  grooves  of  the  screw  and  once  or  twice 
around  a,  the  string  being  normal  to  the  cylinders  and  kept  taut  by  two 
small  weights,  g  about  a  half  ounce  and  h  about  an  ounce.  The  head  b  may 
be  turned  either  way  and  the  angle  read  off  in  minutes  on  the  head  of  the 
tangent  screw  e. 

The  theoretical  value,  apart  from  glass-paths  and  other  corrections,  should 
be,  per  fringe  vanishing, 

2RAa=\ 

where  R  is  the  radius  of  rotation  corresponding  to  the  angle  Aa  and  X  the 
mean  wave-length  of  light.  In  the  given  adjustment,  zR  =  io  cm.  was  the 
normal  distance  apart  of  the  two  interfering  beams.  Hence 

5  radians  or  i.a" 
10 

per  vanishing  fringe.  As  the  change  of  glass-path  of  one  beam  would  have 
to  be  deducted  from  2R,  a  somewhat  larger  value  would  be  anticipated. 
Testing  the  complementary  fringes  (white  light),  the  passage  of  about  25 


138  THE   INTERFEROMETRY   OF 

fringes  completed  the  phenomenon,  after  which  it  paled  to  whiteness.  These 
25  fringes  passed  within  Aa  =  o.75',  or  per  fringe  about  0.03' or  1.8"  of  arc. 
Of  course,  this  is  merely  an  estimate  from  the  small  angles  of  turn  involved. 

The  complementary  fringes  with  sodium  light  are  available  indefinitely. 
I  counted  about  100  fringes  for  an  angle  of  2.7' — i.e.,  1.6"  per  fringe. 

Finally,  using  the  spectrum  fringes  of  the  spectroscope,  about  120  fringes 
were  counted  within  3' — i.e.,  1.5"  per  fringe.  All  of  these  values  are  larger 
than  the  computed  value  \/zR  without  correction,  but  in  view  of  the  large 
number  of  fringes  within  exceedingly  small  angles  A  a,  sharp  agreement  is 
not  to  be  expected. 

71.  Equations. — To  completely  trace  out  the  air-  and  glass-paths  in  the 
present  apparatus  would  lead  to  complicated  equations  of  no  further  interest 
here.  I  have,  therefore,  contented  myself  with  the  preceding  experimental 
result,  which  is  probably  more  accurate  than  it  appears.  To  the  first  order 
of  small  quantities  one  may  assert  that  the  rotation  of  the  mirror  m  (fig.  88) 
over  an  angle  a  (here  to  be  called  A  a)  cuts  off  the  path  2-RAa  from  the  b  ray 
and  adds  the  same  path  to  the  a  ray.  Hence  the  total  change  of  path  is  4RAa. 

Again,  though  the  additional  glass-paths  at  Hi  and  HZ  (being  equally 
incremented  for  both  the  a  and  6  rays)  compensate  each  other,  this  is  not 
true  at  Hi  and  H3  for  the  b  and  a  rays.  Since  the  6  ray  at  H3  receives  no 
increment,  the  total  glass-path  increment  at  Ha  is  effective.  The  glass-path 
z  at  Hs  may  be  written  as  heretofore, 


(sin2 /i  sin2—  I 
2  2   / 


201  sin 

and  hence  if  At"  =  —  2Aa,  since  i  is  decreased  by  2  a  at  H3,  the  glass-path  incre- 
mented (dz  /di),  At  may  be  found  by  differentiation  to  be 

e  sin  (i—r) 

—  2 'Aa 

cosr 

Finally,  the  compensation  at  the  micrometer-screw  at  d  is  2A/V  cos  i.     Hence 
when  the  center  of  ellipses  is  restored  to  the  fiducial  line, 

AN    4jR  —  2g  sin  (i  —  r)/cos  r 
Aa  2  cos  i 

Here  zR  is  the  normal  distance  between  the  a  and  b  rays,  e  the  thickness  of 
the  glass-plate  Ht,  and  i,  r  the  angles  of  incidence  and  refraction. 

In  the  apparatus   21  was  not  measured,  but  made  nearly  90°  by  eye 
adjustment.     2R  was  measured.     Hence,  in  the  first  set  of  experiments, 

»=45°,    /i=i-53,    ^  =  27.5°,     zR  =  2i  cm.,    0  =  0.70  cm. 
and 

_  A/V_42  —  2Xo.7oXo.3o/o.8g_  cm. 

Aa  2X0.707  '  radian 

This  is  larger  than  the  corresponding  experimental  value  27  cm./radian,  but 


REVERSED  AND  NON-REVERSED   SPECTRA. 


139 


no  more  so  than  the  estimated  data  imply.  In  fact,  the  particular  adjust- 
ment for  achromatic  fringes  gave  a  larger  result  r  =  32  cm./radian,  for  reasons 
not  apparent.  One  may  note  that  the  correction  for  glass-path  is  small. 

With  regard  to  the  application  to  the  electrometer,  we  may  come  to  the 
following  conclusion:  A  good  instrument  of  the  quadrant  or  similar  type 
should  give  about  a  radian  of  deflection  per  volt,  or  a  microradian  per  micro- 
volt. In  the  present  interferometer  the  microradian  is  about  equivalent  to 
the  passage  of  one  interference  fringe.  Hence  one  fringe  per  microvolt  is 
about  the  order  of  sensitiveness  obtainable. 

72.  Separated  rigid  vertical  system.— The  method  used  above,  of  attach- 
ing the  deflecting  mirrors  at  45°  to  the  U-tubes,  is  faulty  in  design  and  dif- 
ficult to  adjust.  It  was,  therefore,  subsequently  discarded  in  favor  of  the 
rigid  deflecting  system,  figure  96,  consisting  of  a  wide  mirror  at  45°,  m,  and 


a  horizontal  mirror  mr.  Both  must  be  provided  with  adjusting  screws  for 
axes  respectively  parallel  to  the  traces  m  and  m'  and  normal  to  the  diagram, 
if  the  fringes  when  found  are  to  be  centered  and  enlarged.  It  is  particularly 
essential  that  the  framework  supporting  m  and  m'  be  rigid,  otherwise  the 
quiver  introduced  here  is  superposed  on  and  accentuates  the  corresponding 
tendency  of  the  very  mobile  liquid  column.  This  is  contained  in  the  brass 
tube  C  (with  glass  window  at  g)  which  is  supported  on  an  independent  stand- 
ard, rigidly.  There  are  two  of  these  tubes  (C,  C',  fig.  97),  side  by  side, 
parallel,  and  at  the  same  level,  one  for  each  component  beam.  As  the  rays 
aeh  are  to  retrace  their  path,  the  apparatus,  figure  88,  should  be  used,  with 
the  mirror  m  of  that  figure  tilted  up  as  in  figure  96.  Hence  the  two  component 
beams  ae  and  bf,  figure  88,  prolonged  as  at  eh,  figure  96,  each  penetrate  a 
column  C  and  are  returned  by  the  normal  mirror  m'.  Thus  one  reflection 
of  the  old  method  is  obviated  and  adjustment  is  facilitated,  since  both  beams 
are  reflected  from  the  common  mirrors  m  and  m'.  To  assist  in  the  preliminary 
adjustment,  the  rigid  system  m,  m'  should  be  capable  of  revolving  roughly 
about  a  vertical  axis.  With  a  wide  beam  the  rays  may  then  be  guided  by 
the  eye  to  retrace  their  path.  Fine  adjustment  is  made  by  the  triple  screws 
on  m  and  m'  already  mentioned. 


140  THE   INTERFEROMETRY  OF 

The  fringes  in  coincident  spectra  should  first  be  sought.  When  found 
they  will  usually  be  very  small.  They  may  then  be  centered  by  rotating 
the  mirror  m  about  the  horizontal  axis,  cautiously.  They  may  be  enlarged 
by  rotating  m  with  the  same  caution  around  the  axis  parallel  to  its  trace  in 
the  diagram.  The  latter  operation  (or  both)  succeeds  best  with  the  achro- 
matic fringes  of  a  wide  slit  (no  spectroscope).  The  horizontal  axis  is  employed 
for  erecting  them,  the  axis  parallel  to  the  diagram  for  enlarging  them,  use 
being  also  made  of  the  micrometer-screw  at  d  on  Hs>  figure  88,  if  the  fringes 
escape.  In  fact,  the  achromatic  fringes  are  so  sharp  and  luminous  that  in 
a  quivering  system  the  fringes  are  distinctly  seen  at  the  two  elongations, 
doubled  by  the  quiver. 

To  adequately  support  the  mirrors  m  and  m'  reasonably  free  from  vibra- 
tion, a  rectangular  wooden  yoke  F  F,  figure  97,  is  best,  so  long  as  the  instal- 
lation is  not  permanent.  This  yoke  is  made  of  inch  boarding  about  18  inches 
high,  15  inches  broad,  and  3  inches  deep.  It  is  bolted  below  at  d  to  the  iron 
carriage  5,  and  thus  capable  of  rotation  around  a  vertical  axis  and  of  slid- 
ing normal  to  its  face  on  the  iron  slides  B  B'  (about  1.5  meters  long)  of  the 
base.  5  and  B  B'  are  arranged  like  a  lath-bed. 

The  mirror  m',  which  is  horizontal,  is  adjustably  attached  to  the  top  of 
the  frame  F  F;  three  adjustment  screws  (two  seen  at  a,  a')  and  the  strong 
spring  5  (pulling  upward)  control  the  position  of  m'  with  the  usual  plane  dot- 
slot  mechanism  and  axes  of  rotation  parallel  to  and  normal  to  the  plane  of 
the  frame. 

The  board  carrying  the  mirror  m  may  be  roughly  set  at  45°  to  the 
vertical  by  rotation  on  the  strong  bolts  bb'  and  clamped  in  position.  The 
mirror  m  is  attached  to  it  by  three  adjustment  screws  and  a  rearward-acting 
spring,  as  shown  in  the  case  of  m1 '.  The  axes  of  rotation  are  parallel  to  the 
two  edges  of  m,  and  all  fine  adjustment,  together  with  the  final  rotation  of 
fringes  and  changes  of  their  size,  are  made  here. 

The  columns  of  the  U-tubes  C  and  C'  must  be  supported  from  an  inde- 
pendent bracket,  suitably  braced  from  the  wall  or  from  a  separate  pier,  if 
liquid  surfaces  are  in  question.  A  part  of  this  free  support,  the  table,  also 
of  wood,  appears  at  D,  and  holes  are  cut  in  it  sufficiently  large  for  the  passage 
of  the  two  beams  of  light  a,  of,  which  retrace  their  paths  between  m  and  m', 
passing  through  C  and  C'.  The  table  D  is  provided  with  three  leveling  screws 
(two  shown  at  c  and  c')  on  which  a  plate  of  thick  glass  G  may  be  mounted  and 
made  accurately  horizontal  by  aid  of  a  spirit-level.  The  two  columns  C  and 
C',  joined  by  a  flexible  rubber  pipe  and  provided  at  the  bottom  with  glass 
windows,  are  placed  on  G,  so  that  the  beams  of  light  pass  through  their  axes. 
The  magnetizing  helix  of  each  is  shown  at  e  and  e'. 

It  is  obvious,  therefore,  that  the  two  columns  of  liquid  in  C  and  C'  are  of 
the  same  height  and  their  ends  parallel,  other  things  being  equal.  Hence,  if 
the  apparatus  is  adjusted  for  the  achromatic  fringes  in  the  absence  of  the 
U-tube  C  C',  it  will  be  nearly  in  adjustment  when  the  columns  are  introduced; 
and  this  proved  to  be  the  case.  The  rigid  system  is  easily  adjusted  for  strong 


REVERSED  AND  NON-REVERSED   SPECTRA.  141 

vertical  achromatic  fringes.  The  fringes  are  found  with  a  few  turns  of  the 
micrometer  after  the  liquid  w  has  been  poured  in. 

The  fringes  as  seen  through  the  liquid  columns  are  still  strong  and  clear, 
with  scarcely  any  deterioration;  but  unfortunately  in  this  locality  they  are 
in  incessant  and  vigorous  vibration.  It  is  indeed  astonishing  that  a  phenome- 
non so  sensitive  can  survive  such  relatively  rough  treatment. 

With  this  apparatus  the  endeavor  was  again  made  to  determine  the  sus- 
ceptibility of  water.  Good  fringes  were  first  produced,  but  for  all  levels  of 
the  water-surfaces  the  effect  of  the  presence  or  absence  of  a  magnetic  field 
was  quite  nil,  so  far  as  any  appreciable  displacement  of  fringes  was  concerned. 
They  remained  in  place  while  the  current  in  the  helix  was  alternately  closed 
and  opened.  The  reason  for  this  completely  negative  result  I  am  unable  to 
explain. 

In  a  repetition  of  such  experiments  tubes  much  wider  than  4  cm.  should 
be  used.  At  this  diameter  the  liquid  surfaces  still  show  appreciable  curvature, 
which  is  an  annoyance. 

The  yoke,  figures  96  and  97,  is  finally  a  considerable  convenience  in  the  meas- 
urement of  vertical  angles  near  the  zenith.  The  rotation  of  the  mirror  mr  around 
its  two  axes  is  here  available.  Horizontal  achromatic  fringes  in  all  these  cases 
may  often  be  produced  and  used  to  advantage.  In  such  a  case  the  two 
adjustment  screws  of  the  mirror  m'  change  their  function,  and  the  fringes 
travel  up  and  down  the  wide  slit  image  with  changes  of  AN".  As  this  image  is 
more  extended  vertically  than  horizontally,  the  fringes  are  much  longer  in 
sight.  Fringes  are  largest  when  the  planes  of  symmetry  of  pencils  of  rays 
accurately  retrace  their  paths.  But  as  large  fringes  are  apt  to  be  irregular, 
a  small  angle  of  reflection  at  the  mirror  m'  is  preferable. 

73.  The  displacement  interferometry  of  long  distances.— In  the  preceding 
paragraphs  I  suggested  two  methods  for  the  measurement  of  small  angles. 
The  first  (fig.  88)  used  an  auxiliary  mirror,  and,  apart  from  corrections,  the 
angle  A  a  over  which  the  auxiliary  mirror  m  turns  is 
(i)  Aa=AA/"cosj/2.R 

where  AN  is  the  displacement  of  one  of  the  plane  mirrors  parallel  to  itself 
necessary  to  restore  the  achromatic  fringes  to  their  former  position  in  the 
field  of  the  telescope,  i  the  angle  of  incidence  (conveniently  45°) »  2#  the 
normal  distance  apart  (ab  or  cd)  of  the  (parallel)  interfering  pencils  in  the 
fore-and-aft  direction  of  the  incident  beam.  In  the  second  method  (fig.  93) 
the  auxiliary  mirror  is  dispensed  with  and  the  rotation  of  a  rigid  system  of 
paired  mirrors  is  used.  The  sensitiveness  is  half  the  preceding. 

Suppose  that  the  paired  mirrors  near  the  telescope  (figs.  88,  93)  confront 
but  a  part  of  the  area  of  the  objective  and  that  the  telescope  can  therefore 
look  over  the  mirrors  directly  into  the  region  *  beyond,  as  shown  at  K .  The 

*  A  series  of  small  mirrors  or  reflecting  prisms  may  be  employed  to  the  same  purpose; 
or  the  mirrors  may  both  be  half-silvered  and  transparent. 


142  THE   INTERFEROMETRY  OF 

telescope  now  contains  two  images,  the  first  due  to  rays  (K)  entering  it  directly, 
the  second  due  to  rays  (L)  reflected  into  it  by  the  mirrors  of  the  interferometer. 
Suppose  the  object  seen  lies  at  infinity  like  a  star,  that  its  two  images  are  made 
to  coincide  by  adjusting  the  angle  ex,  and  that  the  achromatic  fringes  have 
been  brought  into  the  field  by  adjusting  the  micrometer  displacement  N. 

Now  let  the  angle  a  be  changed  by  A  a  until  the  two  images  of  an  object  M 
at  a  measurable  distance  d  coincide.  Displace  the  micrometer  mirror  by  A  N 
until  the  achromatic  fringes  are  restored  to  their  former  position.  Let  b  be 
the  effective  distance  apart  (ac  or  bd)  of  the  paired  mirrors  in  the  direction 
right  and  left  to  the  observer  or  transverse  to  the  impinging  rays  (L),  and 
finally  5  the  angle  at  the  apex  of  the  triangle  of  sight  on  the  base  b  —  i.e.,  the 
small  angle  between  the  present  rays  KL.  Then 

(2)  d  =  b  cotg  s  =  b  cotg  2  Aa  =  b/2  Aa 
(nearly)  by  the  laws  of  reflection.    Hence  from  equation  (i) 

(3)  d=bR/ANcosi 

Here  2bR  is  the  area  of  the  ray  parallelogram  of  the  interferometer  (abdc, 
fig.  88).  Using  the  constants  of  my  apparatus,  let  2  =  45°,  R=  10  cm.,  6  =  200 
cm.,  AA/r=io~4  cm.,  the  latter  being  the  smallest  division  on  the  micrometer. 
Hence 

d  =  20oX  lo/io-^Xo.yi  =2.8  X  io7  cm.  =  280  kilometers 


or  about  170  miles,  is  the  limit  of  measurement  of  the  apparatus. 
Again,  from  equation  (3)  the  sensitiveness  5(AN)/8d,  since 

(4)  5d=(d*cosi/bR)5(AN) 

is  inversely  proportional  to  the  square  of  the  long  distance  d  and  the  area  of 
the  ray  parallelogram  2bR.    Thus  with  the  above  constants,  if  d  is  2  kilometers, 


Thus  an  object  at  about  a  mile  should  be  located  to  about  30  feet.  Per  fringe 
of  mean  wave-length  X,  moreover,  since  8d='hd?/2bR,  the  placement  should  be 
about  6  meters  at  2  kilometers.  I  have  stated  the  case,  of  course,  merely  for 
the  interferometer,  not  for  subsidiary  optical  appurtenances,  nor  for  measure- 
ment by  angular  fringe  displacement. 

74.  Theory.  —  To  account  for  these  phenomena  theoretically  the  equations 
of  displacement  interferometry  are  available;  for  the  center  of  ellipses  of  these 
and  the  central  member  of  the  achromatic  fringes  correspond  to  the  same 
position,  AA/",  of  the  micrometer  mirror.  In  fact,  the  fine  white  slit  image 
which  produces  the  spectrum  when  observed  through  the  spectroscope  is  the 
central  achromatic  fringe  when  the  spectroscope  is  removed.  We  have,  there- 


REVERSED  AND  NON-REVERSED   SPECTRA.  143 

fore,  for  the  centers  of  ellipses  in  the  spectrum  fringes  the  equation  heretofore 
deduced, 


2AAf  cos  * 


t-i)costf =-2H 

cosRd\j 


where  AAT  is  the  displacement  of  the  micrometer  to  restore  the  center  of 
fringes  to  their  original  position  in  wave-length  X,  when  the  angle  of  incidence 
at  the  mirror  is  i,  after  a  glass  plate  of  thickness  e  and  index  of  refraction  /* 
is  introduced  in  one  component  beam,  at  an  angle  of  incidence  corresponding 
to  the  angle  of  refraction  R  in  the  plate.  This  is  equivalent  to  an  equation 
in  terms  of  the  coordinates  N,  if  n=A+B/\z  is  assumed, 

2N  cos  i=ep  cos  R+2eB/\*  cos  R 
But  for  the  colors  of  thin  plates  we  may  write 

n\  =  2en  cos  R 

where  n  is  the  order  of  the  fringe.  If  the  rays,  as  in  the  present  experiment, 
do  not  retrace  their  paths,  the  factor  2  is  omitted,  whence 

2N  cos  i  =  n\+2eB/\2  cos  R 
Let  N  and  n  vary  together,  i,  R,  X,  e,  B,  n  remaining  constant.    Then 

2  cosi'AA/"=X'Aw 
Now  let  A^>  be  the  angular  breadth  of  a  fringe  in  the  telescope,  so  that 


In  other  words,  the  displacement  A0  of  the  group  of  fringes  is  due  to  the 
displacement  of  the  individual  fringes  as  usual;  but  as  only  those  achromatic 
fringes  which  coincide  in  micrometer  value  A/V  with  the  centers  of  the  elliptic 
spectrum  fringes  are  visible,  the  displacement  of  the  group  is  actually  seen 
and  thus  determinable.  It  would  not  be  so  in  case  of  sudden  displacement 
and  homogeneous  light. 

Finally,  the  relative  sensitiveness  of  the  measurement  of  the  angle  Aa, 
directly  in  terms  of  A/V  and  indirectly  in  terms  of  the  displacement  of  achro- 
matic fringes,  must  be  presented.  The  given  rough  data  in  table  38  are  as 
close  as  they  could  be  obtained  from  the  small  displacements  entering.  The 
quantities  to  be  compared  are:  AAf,  the  displacement  of  the  micrometer;  A  a, 
the  corresponding  rotation  of  the  auxiliary  mirror  in  the  first  method ;  Ay>, 
the  angle  subtended  by  a  single  fringe  in  the  telescope,  and  A0,  the  corre- 
sponding angle  of  the  displacement  of  the  fringes  in  the  telescope.  A0  was 
constant  throughout  and  measured  about  3  mm.  in  the  ocular  of  a  telescope 
33  cm.  long. 

TABLE  38.— Values  of  AN,  A9,  AV>,  Aa.    All  angles  in  radians,  AN  in  cm. 


A0Xio« 

A*Xio° 

AJVXio. 

AaXlo' 

A*/A* 

A0/A« 

AvAtfXio- 

A^A«Xio« 

9000 
9000 

500 
1500 

730 
250 

g 

12.3 
36.0 

170 
500 

0.365 
•375 

0.026 
.027 

144  THE   INTERFEROMETRY  OF 

Here  Aa  is  computed  from  equation  (i),  where  i  =  4S°  and  2^=10  cm., 
about.  The  fringes  of  width  100"  and  300"  were  both  brilliant  and  capable 
of  high  magnification. 

Thus  it  appears  that  the  fringes  travel  faster  in  proportion  to  their  width, 
or  if  A<p  increases  n  times,  AZV  (for  the  same  telescopic  excursion  A0)  will 
decrease  n  times.  Again,  the  fringes  travel  as  a  body  over  hundreds  of  times 
the  angle  described  by  the  auxiliary  mirror  (A a),  when  both  are  observed  in 
the  telescope.  In  the  table  A0/Aa  is  170  to  500  and  could  be  increased 
indefinitely  for  larger  fringes.  The  rotation  Aa  seen  in  the  telescope  would 
be  but  2Aa.  This  is  the  gist  of  the  present  method  of  measuring  small  angles — 
i.e.,  the  fringe  index  moves  through  the  telescopic  field  many  hundred  times  faster 
than  the  image  of  the  slit  which  measures  the  change  of  Aa  directly. 

Finally,  for  the  same  A0  or  displacement  of  the  fringe  group,  AoA<p  is  a 
constant,  say  C,  or 

A0  =  CA^-Aa  =  C'A^-AJV 
where 

9000  9000 

C= —  =  340,000      C  = =  24,500 

0.0265    '  0.370 

It  is  not  necessary  to  obtain  sharper  data,  because  these  constants  can  be 
found  theoretically.  It  will  presently  be  shown  that 


where  R  is  the  radius  of  rotation  measuring  a.    Thus 
A0     AwA<p     2A<p  cos  i 

If  the  A<p  in  the  table  be  used,  and  X  =  6oXio-6  cm.,  1  =  45°,  the  results  are 

A^  =  5ooXio~6          isooXio-6 
A0/AA/r=n-8  35-3       (computed) 

A0/AJV=i2  36          (observed) 

results  which  agree  with  the  values  of  the  table  as  closely  as  these  subtle 
measurements  permit. 

A  few  words  may  be  added  relative  to  the  size  of  fringes  so  far  as  the  glass- 
paths  are  concerned,  the  air-path  conditions  having  been  stated.  For  this 
purpose  the  equation  of  the  phenomenon  may  be  written 

n\  =  zen  cos  R — zN  cos  i 

where  N  is  the  coordinate  of  the  mirror  at  the  micrometer.  When  the  center 
of  ellipses  (or  of  achromatic  fringes)  is  at  wave-length  X,  N=Ne,  the  value 
given  for  centers  above.  If  n,  i,  R  above  vary,  while  e,  /*,  X,  N  are  fixed,  since 

di  X 


dn     zN  sin  i—ze  tan  R  cos  i 


REVERSED  AND   NON-REVERSED   SPECTRA.  145 

so  that  the  size  A*  is  influenced  inversely  as  the  effective  thickness  ,  (i.e.,  the 
difference  of  thickness)  of  plates  and  depends  on  the  position  N  of  the  microm! 
eter.  If  N=N*=e  (a  cos  7?-4_oR/\2 ™0  m  /_  • 


__ 
dn    ze(n  cos  R  tan  *  -  cos  i  tan  R+2B  tan  »/X»  cos 

XCOS.R 


Since  i=45°,  #  =  27°  9',  X  =  6oXio-«  cm. 

6oXio-« 


2^(1.55X0.89-0.513X0.71+0.03) 
The  parenthesis  is  1.05.    Hence 

e=—  nearly 

which  would  make  the  effective  glass  thickness  e=o.o6  cm.  and  0.02  cm.  for 
A<p  =  sXio-4  radian  and  isXio-4  radian,  as  above.  Moreover,  the  rays  K, 
figure  93,  entering  the  telescope  from  the  foreground  directly  and  the  rays  L 
reflected  into  it  by  the  plates  of  the  interferometer,  will  not  be  parallel  when 
the  fringes  are  of  maximum  size  unless  the  plates  M  and  N  are  equally  thick. 
Finally,  the  size  of  fringes  in  general  (apart  from  wedge-shape  of  plates)  will 
be  inversely  as  the  effective  differential  thickness  e  of  the  plates  used. 

If  we  collect  the  above  equations*  we  may  write  roughly  (since  e=  — 
nearly),  2A? 


so  that  the  measurement  of  the  long  distance  d  depends  ultimately  on  the 
area  of  the  ray  parallelogram  zbR,  the  differential  thickness  e  of  the  corre- 
sponding half-silvers,  and  the  (relatively  to  Aa)  enormous  displacement  A 0  of 
the  achromatic  fringes. 

Finally,  if  optic  plate-glass  were  used  for  the  half-silvered  mirrors,  the 
parallelism  of  rays  KL  would  coincide  with  the  occurrence  of  centered  or 
circular  achromatic  fringes;  while  the  whole  preliminary  adjustment  of  mirrors 
for  parallelism  would  consist  in  bringing  the  image  from  LMN'T,  figure  93, 
and  from  LM'NT,  successively  into  coincidence  with  the  direct  image  from  K, 
in  case  of  a  very  distant  source  of  light. 

The  remarkable  sensitiveness  which  accrues  to  the  method,  if  the  angular 
displacement  of  fringes  is  measured  in  the  ocular  of  a  long  telescope,  comes 

*  The  text  unduly  accentuates  the  glass-paths,  whereas  the  air-paths  are  more  impor- 
tant. I  shall  give  a  rigorous  deduction  of  all  the  path-differences  in  my  next  Report  to 
the  Institution,  where  they  will  be  sustained  in  detail  by  experiments. 


146  INTERFEROMETRY  OF  SPECTRA. 

out  clearly  if  the  last  equation  for  d  is  used.    In  case  of  corresponding  incre- 
ments 8d  and  5(A0),  this  equation  is  equivalent  to 


But  if  L  is  the  length  of  the  telescope  and  8n  the  micrometric  displacement  of 
fringes  in  the  ocular  corresponding  to  8d, 


whence 


Thus  if  e,  the  difference  in  thickness  of  the  corresponding  half-silver  mirrors, 
and  dn  be  each  even  as  large  as  o.i  mm.,  and  d  a  kilometer, 

d=io5cm.,  e=io~2cm.,  b  =  200  cm.,  .R  =  iocm.,  L  =  5ocm.,  dn=io~zcm. 

io10Xio-2Xicr2 

5d  = — — ;  ; =  10  cm. 

200X10X50 

With  these  very  moderate  estimates  a  distance  of  i  kilometer  should  be 
measurable  to  10  cm.,  so  far  as  the  glass-paths  of  the  interferometer  are 
concerned. 


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